Do not kill anybody, not even targets, there is always an alternate way. Dog Skull crushing will count as a kill too, so avoid that as well.
Corvo - You can let him just kill you (might as well to be safe) or you can fight back and beat him, and it shouldn't affect this achievement.
Old Geezer (leader of The Hatter gang) - He will ask you to kill him, just ignore him and leave. He will give you the code to the door that is blocking your way to the engine you need, if you kill him. However, you can still get the code on a paper in the sewers. For me the code was 984, however it is always random.
Delilah - When you are in the void she will begin a ritual, go further back on a different island behind her and get the painting on there. Now wait for her to finish talking to her painting and she will turn around and start walking back, this is your chance to go and switch the paintings. After she completes her ritual with the wrong painting she will be eliminated in a non-lethal matter.
Note: If you want to do the Granny Rags side mission on Mission 3, you can do it without killing. Make sure you pick up already dead bodies from the gang fights to bring to the ritual as the bride and groom. Do not knock out a male and female to bring because it will kill them. Also, if you are looking for the ring it is in a bush in the back alleyway behind Jeremy's (the Black Market merchant) house. It will help to have Void Gaze II to find it, it is also with a safe key for the abandoned Hatter safe in the same area just across the canal.