The challenge location coordinates and the screenshots belong to louster from the UnEpic official forums, for the original post clic here[www.unepicgame.com]
The information on how to complete each challenge was done with my knowledge and some information posted on the UnEpic forums.
Thanks to louster for letting me use his guide and thanks to the people who took their time to record some video guides of this game.

The location coordinates will be given in the form [x, y], where [0, 0] is the central golden box, [-x, -y] is towards the top left of the map, and [+x, +y] is towards the bottom right of the map. The order will be roughly the same as the order you encounter them in-game. Each challenge will also include relevant screenshots, in which I've marked the locations of false walls, etc.
For challenges that are not located within the map proper, the coordinates will be for the on-map location that accesses them. For these I've generally included screenshots of the sub-map's 'entrance' location as well as the route to the challenge.
Use this map if you have trouble following the coordinates:

So what do we get from completing challenges apart from the achievement? We get UNepic Points (UN for short from now on). With these points it's possible to buy weapons, armors, pets and some items in Gualix's shop[unepic.wikia.com]. These points can only be obtained once per challenge.
A total of 750UN can be achieved.
Creating a new character will make him spawn with all your Unepic Points achieved to date, even if you have spent some of them with another character.
The full item list can be found here[www.unepicgame.com].
It's up to you to use these items or play the game as it is.

Description: 'Start a new game and reach the first spirit box unharmed'
Location: Central Zone [1, 0]
Reward: 20UN

How to complete:
Start a new game on any difficulty and don't get hit, if you get hit just restart the game once again, it's not hard to get to the spirit box without getting hit.

Description: 'Do not drink any potions until you have defeated the Dynidon worm'
Location: Central Zone [1, 2]
Reward: 20UN

How to complete:
Don't link any potion to the key binds of your keyboard, that way you won't end up using them accidentally. Remember to play carefully and go to the box each time you need, your goal is to not die so the enemies don't respawn, just try to get to the boss asap and beat him, he's not hard to kill without getting hit more than 1 or 2 times.

Originally posted by Tsunetoma:

If you have enough unepic points (what you get from completing challenges) to buy the mini spaceball (300UN) from the merchant below the first pure spirit. It will automatically kill Dynidon

Description: 'Do not touch the ground or ladders for 18 seconds'
Location: Central Zone [2, -1]
Reward: 20UN

How to complete:
Go to the mines and take the elevator to the top room [6, 1], you have to use the ropes to fall down without touching the floor. This can be done from the first room of the mines [6, 2] too, the one that contains the first gate. It will take a few attempts to do until you memorize the jumps.

-SNED-Unipigdog posted a video guide for this challenge, take a look at it.
http://steamproxy.com/sharedfiles/filedetails/?id=153372952

Description: 'Complete the room of traps unharmed in 50 seconds'
Location: Central Zone [0, 2]
Reward: 20UN

How to complete:
The traps room is above the one where you get the challenge.

Konargus posted a video guide, take a look at it, it explains an easy way to complete the challenge.
http://steamproxy.com/sharedfiles/filedetails/?id=204900206 This challenge doesn't have to be completed when you do the quest of this room, you can return later on, the traps will still be there and the timer will start normally.

This part is probably obsolete, the video guide contains an easier method.

Originally posted by MaximumDestroyer:

You need ghost potions or spells to complete this, they will allow you to dodge everything except fireballs, so remember to crouch or jump to avoid them. Movement speed potions and phase door spells/potions help too.
Bring some extra potions in case you fail, don't try to rush the last part, there's 4 fireballs and it's better to dodge 1 or 2 of them at the same time.

Description: 'Dodge all falling rocks between the columns in the corridor to the mine'
Location: The Mine [6, 2]
Reward: 30UN

How to complete:
Wait until the rocks do only 1 damage to you to attempt this challenge.
This challenge needs some luck and good reflexes to complete, surviving 1 minute there is no easy task. Speed potions are really usefull here, the extra movement speed helps a lot.
Using ghost potions and ghost spells will remove the timer, you can't use them.
Rocks seem to fall at a faster speed when there's 15 seconds left.

Konargus posted a guide for this, he suggests the use of the pet Gatchan to paralyze the rocks.
http://steamproxy.com/sharedfiles/filedetails/?id=204562221

Originally posted by Ignisterra:

If you didn't use the melee immunity scrolls, they can be very very useful at the end when the rocks fall quicker at 15 seconds.
The rocks won't damage you and it will not deactivate the timer.
I've found only two of them, so it granted me between 15 and 20 seconds of GOD MODE.
Don't forget the casting time.

Originally posted by Dave:

An easy way to get the "King of Dodge" is to just barely go far enough into the mine to start the timer. Very few of the rocks will fall near you, and those that do will all come from the same direction so they're easy to see coming.

Description: 'Carry a collection of at least 18 different potions at once'
Location: West Zone [-3, -1]
Reward: 80UN

How to complete:
By the time you get to the end of the game you should be able to craft 17 potions, some potions like advanced healing potion and advanced regen potion can't be crafted keep at least one of those in your inventory. Remember to have enough space in your inventory, since you need to have the 18 potions at the same time with you.

Description: 'Find Harnakon without killing any crows beforehand'
Location: The Library [6, -8]
Reward: 60UN
Entrance of the challenge:
Inside the door:
If you are having a hard time breaking the wall try switching to a faster weapon.

How to complete:
This challenge will be hard to do, be carefull with your pets, if they kill a crow you won't be able to get the achievement as that counts as if you killed them yourself.
Remember not to carry any ring when you enter a room with crows, or you won't be able to get it back if you want this achievement, you can store them on Harnabank.
Check this guide by BISTOR to know which rooms contain crows.
http://steamproxy.com/sharedfiles/filedetails/?id=260938319

Description: 'Carry 666 essences of each color, 666 coins and 666 divine favors at the same time'
Location: The Catacombs [-5, 1]
Reward: 80UN

How to complete:
Use your extra money to buy essences on the kitchen.
To farm divine favors fight skeletons with the Paladin's Hammer or Light Salvation spell. If you can use light robes use them as well.
You need to carry exactly 666 of each one of those, you will need to use some money and drop essences after farming them in order to get that exact number.

Description: 'Defeat Sux Mortis with skill level 1 in armor and constitution'
Location: The Catacombs [-7, 3]
Reward: 50UN
The entrance is the Mausoleum Magnum:
The location is the 2nd area of the Mausoleum Magnum, at the bottom, you have to break the wall while falling:

How to complete:
This won't be an easy task, you cant put any skill point on armor and constitution, that means that your HP will be really low and your defenses too.
You will have to play carefully and carry lots of healing stuff, there's no tricks here, you need to show your skills to get this achievement.

Here's some help posted by other people:

Originally posted by RavenForce:

There's an easier way to beat Death..I mean Sux Mortis(I checked and mortis=death in latin!).You can either farm skeletons in the catacombs/top garden area to fight the boss with light bolts,or use fire spell of Fireball/Improved firebolt to fight from far.On ground you want to prepare either hammer/axe to deal the boss and/or deal with the skeletons.As for the skeletons,they're dealt in melee,though in Hard++ it's almost impossible to deal so many with your weapon,you may want to use fire breath.Again,with polymorph wand it's easy in case arcane/fire aren't in your spells.Light pet's optional here,from experience.To avoid the attacks,when he's above,learn the right timing to move around to avoid of dying.In one of the ground attacks,your best bet's either you're on the right and phase door,or figure out the pattern(it's easy when you gettin' hang of it)

Originally posted by Ash:

There's a dirt cheap trick to complete "Death lvl 1" challenge even on Hard++. Play as usual, get to the catacombs, get magic of light (and some divine favours, around 50), Orb of Rebirth and 4-5 potions of ghost shape. Use Orb - this will reset your Armour and Const skills to 1. Save, go to Sux's chamber, summon him and drink ghost potion after that. Voila - you're invincible to his scythes, you can spam Light bolt or any other spell at him or skeletons, you can drink another ghost potion while you're ethereal. The only attack which can harm you is arkane ground slam, but it can easily be avoided via jump. Spirit of Light greatly helps here, but he's not mandatory.

Description: 'Defeat Gauntlik while carrying 8 vampires in your inventory'
Location: The Prison [-4, -8], leading to [-5, -8]
Reward: 40UN
Breaking this wall will lead you to a new area of the castle, just above the bridge:

How to complete:
The 8 vampires will drain your HP to 1 point, you can't die from their damage, however, getting hit once by the boss will finish you.

Eatitup posted a video guide for this challenge, take a look at it. http://steamproxy.com/sharedfiles/filedetails/?id=153374934

Description: 'Bring 4 or more freed goblins to the mine without using teleport or gates'
Location: The Prison [-4, -5]
Reward: 60UN
Entrance:
Location:

How to complete:
This one is, in my opinion, the hardest challenge of the game, you can rescue up to 8 goblins in this mission, and have to carry 4 of them alive to the mines, this will prove to be really hard because you can't use teleports or gates.
Goblins run really slow, just clear their way of enemies and rush to the other rooms before they get hit, they will appear instantly near you even if they were on the other side of the previous room.

Originally posted by "Ignisterra":

There are 3 things to clearly avoid:
- The Flying helmet in the prison, you have to kill them before freeing the goblins as they will try to attack it and die of burning.

- The spiders in the Gardens, you or the goblins MUST NOT kill them as they will release the poison cloud which will kill all the goblins unless you can heal them until the poison effects disappear. The best is using invisibility + speed up to rush to the exit and run under the starry sky between the East Zone and the Gardens as if the spider see you it will rush to you and the goblins will kill it and die before you can access the exit.

- The falling rocks from the mine, here, ghost form and speed up to run straight to the mine. The goblins shouldn't even be hit by one rock.

Originally posted by "Ulithium_Dragon":

1) YOU HAVE TO FINISH THE QUEST FIRST in order to do this. Once you turn it in, a line of dialog will trigger between you and Zera (sp?), and THEN you can take them to the mine.
2) It's VERY helpful to have the healing spell "heal allies", which requires healing level 5.
3) Having at least one skeleton helps (more cannon fodder to take the hits), as does having a pet out.
4) Having the hidden wall shortcut in the healing box room (found one help up on the right wall) helps a lot because you can skip the room with the vampire bats.
5) The best way to get past the rocks is to simply use a speed potion and a potion of ghost shape/scroll of melee immunity and pray you can run through the room before the goblins get hit. If you get far enough ahead of them, they should stop and display the ! above their head for a second or two. Also DO NOT attempt to stop and heal the goblins in the falling rock room, since there are no completely "safe" areas after the first room.
6) Saving DOES NOT impede this challenge's progress, nor does reloading.

***

***I don't know if it's required or not, but the one time I had to reload I force quit the game, killing the process before it had time to close itself properly. Since simply using alt+F4 only ASKS the program to close, it may give it enough time to write the data marking the character's progress as "failed". Anyways, true force closing can be done either by stopping the progress in the task manager (windows), or by using the applet "Super F4" (which is what I used).

Description: 'Defeat Medeox without killing any of its snakes'
Location: The Halls [7, 0], leading to [7, -1]
Reward: 30UN
This challenge is located near challenge 13, go to that section to see the screenshots.

How to complete:
The easiest way to beat him is to use bows along with some powerups, Giant Strength and Bow Mastery, if you are low on items save them for the 2nd part of the fight, when he drops to 1/2 or 1/3 HP.
If you don't use bows you can kill him using the Transmute Metal spell, but it will take longer than with bows.

Eatitup posted a video guide for this challenge, take a look at it.
http://steamproxy.com/sharedfiles/filedetails/?id=153408693
Zeropoint created a video and an explanation on how to do this challenge:

Originally posted by Zeropoint:

A little trick from zeropoint for "eye for an eye" challenge.
hard++
http://youtu.be/sw2x0BWIag8 i used :
An enchanted "Meltroth" bow (120%) <requires level 12 of bow skill>
Gatchan pet to stun ghosts
1x potion of speed
1x scroll of bow mastery
2x potion of giant strength
3x skeletons (in fact, 2 are enough) <needs at least level 5 of arcane>

Description: 'Use the toy hammer to deal the final blow to a walking armor'
Location: The Halls [7, 0], leading to [6, 0]
Reward: 30UN
The entrance is located inside the room that contains the burocracy quest:
There's another secret wall inside, leading to a door that contains challenge 12 and another exit on the left containing challenge 13

How to complete:
Eatitup posted a video guide for this challenge, take a look at it.
http://steamproxy.com/sharedfiles/filedetails/?id=153621974#97201 In this part of the castle there's enemies that look like jesters, they will cast a spell that transforms your current weapon in a toy hammer that deals 1 point of damage.
Once you have one of these weapons get to the first gate of this area, which contains 2 walking armors. Remember to use the pet Gatchan to paralyze your enemies.
First lower the HP of the walking armor with a few hits, leaving only one to kill it, then swap to your toy hammer and start hitting him, it will take a while to finish the enemy.

Description: 'Kill Corvax the Mage in HARD or HARD++ mode'
Location: Final area, accessed via the Mage Tower [0, -9]
Reward: 60UN
While walking through this area you will have noticed a challenge on the left side, apparently unreachable from your position, you have to get to a higher room and jump to the challenge platform:

How to complete:
As the description says, get to the final area of the game on HARD or HARD++ and beat the miniboss Corvax the Mage, he doesn't have much HP but he can be really annoying with his AoE stun.
Use the Arcane Inmunity scroll to avoid getting stunned.
If you kill Corvax with Acid/Poison you won't get the achievement, this is currently bugged.

Originally posted by >f7> Click Fire | Nox:

a note I found for die hard, I used a spell of reflect projectiles and stood inside him as he shot, it instant killed him and gave me credit. So yeah, die hard is actually fairly easy.

Description: 'Reach the Twin Crystals in HARD++ mode'
Location: Final area, accessed via the Mage Tower [0, -9]
Reward: 150UN
Get to the final room of this zone and jump from this point:

How to complete:
As the description says you have to beat the game on HARD++ to get the achievement.