You must simply use nonlethal takedowns or avoidance to deal with regular enemies and use the nonlethal neutralisations of primary targets to complete a level.
Things that cause people to die, even indirectly, will spoil the achievement for you, so keep an eye on (e.g.) enemies falling to their deaths if you sleep-dart them. Bloatflies etc. shouldn't count.
It's a good habit to keep an eye on the Stats page that can be accessed from the menu screen before you save. As long as you're keeping at 0 kills, you're fine. If you somehow manage to kill something, revert to previous save.