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This achievement will require that you complete an entire playthrough without ever having killed a single individual. During the first mission in Dubai, make sure you choose the "non-lethal" option when asked how you'd like to approach the situation. Here, you can pick between a Tranquilizer Rifle or Stun Gun.

During this opening mission, and moving forward throughout the game, you must be careful not to kill your enemies. There a number of gadgets and equipment items to deal with hostiles non-lethally (Tranq Rifle, Stun Gun, Gas Grenade, etc.), as well as the traditional non-lethal melee takedown. In my personal opinion, it's best to avoid contact altogether and simply sneak your way around enemies during missions. In addition, this is best stacked with "Foxiest of the Hounds." If you do happen to accidentally kill an enemy, you can reload your latest save and carry on.

Here a few additional tips for this achievement, provided by PowerPyx.

Whenever you enter combat, there is a risk that enemies will throw frag grenades and kill one of their allies. If they die it voids the achievement.
Shooting or triggering explosive traps can kill nearby enemies even when they are knocked out. If something explodes make sure no enemies were affected.
Make sure to take extra care with knocked-out enemies. They can still die if you use something lethal near them.
Enemies that are down will either show the "ZZZ" icon when they are sleeping or a red "skull" when they are dead.
There is one unavoidable death; Talos Rucker will always die in Mission 7, regardless of your dialogue choices. This doesn't affect the achievement, though.
Story character deaths don't matter. Your decisions will affect who gets to live and who dies (the same applies for optional mission objectives). But, because you won't be killing these people directly, it doesn't affect they achievement.
Using Marchenko's kill switch for the "Laputan Machine" achievement will kill him and void the achievement.
In order to take out Marchenko non-lethally towards the end of the game, you can use an EMP Mine to stun him. Once stunned, simply approach him and perform a non-lethal melee takedown. You can refer to the video below for visual guidance.