I'm not totally sure what is required to achieve this but I will state the bits I know and what I believe, the bits I'm unsure of are in underlined.
NOTE 1: As confirmed by Adibou you can use the kill switch to gain this achievement instead of the bomb disarm thus allowing you to choose if you want to do Mission 11 or Mission 12.
NOTE 2: As M33TSPINNER confirmed you can just knock Marchenko out without using the disarm device.
During the last mission once you reach the catering room and had the cut scene the game will tell you to choose between stopping Marchenko or rescuing Brown, choose rescue Brown and make your way to the objective. If you chose to do M12 make sure to give Miller the antidote after the cut-scene to gain another achievement.
Once you have saved Brown you can then move on to the objective of confronting Marchenko. When you meet him during the cut scene you will be given a prompt to use the bomb disarm device or the kill switch, use it and your done. If you have none of these then either use an EMP mine or grenade on him and then go up to him and knock him out.
The achievement should pop once the cut scene before the credits loads.