Start the game on Casual, as it's the easiest mode to snag the achievement on.

Use These Genes
-ATP Boost
-Teracyte
-Ion Surge
-Together Strong
-Extremophile

Start in India
-First Evolve Simian Neuro-genesis.

When you have enough DNA, evolve;
-Primal Awareness
-Ape Colonies
-Organised Travel.

Next, evolve only these transmissions;
-Sebaceous Excretion
-Blood Gift
-Cutaneous Transfer
-Inter-genus Dissemination 1
-Inter-genus Dissemination 2
-Submissive Conception
-Excrement

Now, go back to Abilities and evolve;
-Logistics
-Seafaring

That's enough evolving for now. All you should be worrying about at this point is stopping symptoms from evolving. You should also be establishing Ape Colonies. In my game, I ended up having six.

-It's a good idea to make some save files throughout the game so you don't have to start over from the beginning if something goes wrong.

-Devolve symptoms before closing the pop-up window.
-Save up your DNA. Always make sure you have enough DNA to devolve any symptoms that pop up.
-Most of the DNA you receive will come from your colonies. That's why you'll want to have at least six of them, as devolving symptoms becomes very expensive as the game goes on.

-There are two options for going about the symptoms. The first is to not let any evolve, but this strategy requires you to wait a very long time for the cure to reach 100%. The second option is to evolve only Nausea and Rash. These will get the cure up and running without making the disease's severity increase too much.
-If you do decide to go with the Nausea/Rash route, don't evolve them until at least 99% of apes are intelligent.
Note: I did have the Headache Symptom for a short time without realizing it, but I don't think that it affected the game.

-Keep four of the six Ape Colonies in Asia/Europe/Africa, one in the USA, and one in Brazil. I'm unsure if this is required, but it's how I had the colonies positioned.
-I made 99% of the world's apes intelligent in my playthrough. Enlightening all of those apes requires you to utilize your Organized Travel ability by moving your intelligent apes from country to country, expanding your pack of primates with each migration. This is also how you will establish ape colonies in the required countries, and by doing so, the apes in the surrounding countries will become intelligent as well.
-Island countries typically don't have many apes, but feel free to use Organized Travel to infect them too.

-Don't let too many apes die. If enough apes are killed by the Gen-Sys Labs, the achievement won't unlock. The best way to avoid this outcome is by moving all apes in the country to one without a lab.
-Don't even think about evolving the Ape Rampage ability, let alone using it.

-The cure has to reach 100%, so don't waste DNA on abilities other than the ones listed above.
-You will be defeated, but it's required to unlock the achievement.

-This process takes a long time, so be patient.

-If you've made enough apes intelligent and kept them alive up to this point, you're very close to victory.
-The cure will reach 100%, which is a good thing as long as it only cures the humans(this will only happen if you followed this guide and didn't make the virus a threat).
-Keep in mind that this strategy could take a couple of tries. I was lucky enough to get the achievement on my first attempt, but that doesn't mean it'll go the same way for you.
-If you've followed this guide correctly, a pop-up will eventually appear announcing that humans and apes are to co-exist, triggering the achievement and signaling the defeat of your plague.

Victory Screenshot