For this achievement, you can only tranquilize or knock out enemies. The Mk22, Mosin Nagant, EZ Gun, stun grenades, and CQC are your main tools.
Doing this on easy or very easy will give the EZ Gun, a tranquilizer with infinite ammo, a suppressor that never degrades, infinite ammo, and just holding it boosts stamina and camo index. This alone should be enough to get you through the game. On higher difficulties, this must be unlocked by achieving the rank of Markhor, or in this case the achievement King of the Jungle.
Another item that would help, assuming you have it already, is the stealth item awarded for shooting all Kerotan frogs in one playthrough. Between the stealth item and the EZ Gun, a playthrough can be completed on Normal in about 3-4 hours.
It's been confirmed that you have to stamina-kill Volgin when he's riding the Shagohod, even though there's no camo reward for it. It has also been confirmed that you can kill The End by waiting a week.
In the motorcycle chase scenes, it is acceptable to tranq the soldiers chasing you, even if doing so sends them flying off a motorcycle at top speed to their inevitable death.
If you do end up killing someone in an odd fashion, for instance punching someone out and they drown in water, keep in mind that any non-scripted death is unconfirmed. I'd recommend restarting a save in that case.