Disclaimer: This is very much based off of this Steam forums post. I didn't write it, but I've tried to expand it a little, based on my own playthrough to get the achievement.
First of all, you do need to beat The Rise of Africa before you can do this. It shouldn't take you all that long, and you'll get a (very!) basic feel for how the game works.
After that, play the Fuel Crisis scenario. Start by recruiting a single agent in all regions except Africa. Create a Regional Technology Office in each location, and end the turn.
On the second turn, replace agents that have gone missing, add an agent in both Africa regions, and give them a tech office too. Set Japan to research Energy, and create a Political Office everywhere else. End the turn.
Keep Japan on Energy research until they discover CCS. Everywhere else, try to keep your hearts up so they don't ban you too early, and when you don't have anything else to do in a region, have them commit to nuclear energy. You will also want to play the Black Ops card in the more stable parts of the world; this will help you later.
When Japan gets CCS, have everyone else acquire it. Keep Japan on research.
The next turn, you should get 4th generation nuclear energy in a couple of regions. For those regions who have not gotten there on their own, acquire it.
From here on, simply commit to nuclear everywhere. In the more stable regions, play Fund Insurgency Campaign to reduce stability, then commit to nuclear. Keep doing this, and hopefully, you will quickly reach a global thermonuclear war.
If you fail, try again - there's a fair bit of RNG involved in this game, so you might get banned too quickly, or miss the target before you cause enough instability.
TL;DR: Get tech offices for everyone, have Japan research energy, import CCS and 4th gen nuclear everywhere, commit to nuclear and reduce stability.
Best Answer
There's an eHow article outlining one method (however you need a level 50 character to utilize it):