I do not want to spoil anything here but in order to start this mission, you must complete the "Best Left Forgotten" quest far enough to where it gives you the option to uncover the location of the nuclear launch key. This objective will pop up after completing some of the memory hacks (I really suggest doing all 5 back-to-back). Once it gives you the objective, head into the hotel where the key is located, and the "Cleansing the Land" quest will trigger. Follow it until the end and you get the achievement.
*** Spoiler - click to reveal ***Make sure that you activate the bomb!!!! Talking to the High Confessor and giving him the key will not pop the achievement. (I suggest talking to the High Confessor before starting the bomb just to get the cool perk )