It is possible to get this achievement and all the other ones - which involve a lot of killing of 'innocent' citizens - in a single(-ish) playthrough by backing up your savegame at particular moments in the story.
So play the game without letting anyone die as described in other guides and the walkthrough of the Xbox One version of the game (credits to the writers) and backup your save as described below. After your first playthrough which should unlock the Not Even Once achievement, you can import your backup save to finish the other achievements without restarting the game. If you follow the walkthrough, the 'second' playthrough to get the remaining achievements would be 1 or 2 hours long.
Link to walkthrough: Vampyr WalkthroughEdit: Due to feedback (thanks to CrysisWarXX3) I have included a summary of the choices you'll need to make in the game to unlock the achievement. Again for more information please read the excellent walkthrough.
How to backup and import your saves
When to backup your saves
1) Chapter 1: Take me to the Hospital
There is a point early in the game where you have the choice to embrace (kill) or release Clay Cox. Both will earn you a separate achievement, but for the Not Even Once achievement you obviously have to release Cox. So make a backup right before you make this choice, so you can go back and embrace him without going through the prologue again.
2) Chapter 6: Antidote
Late in the game you have to find ingredients for an antidote. After this mission your next waypoint is towards the sewers and the final boss fight. This is the point of no return. Don't follow this waypoint but backup your save.
There is a lot missable in this game, so I would suggest making a habit of backing up your saves multiple times during the story. If something goes wrong, you have a safety net and you don't have to start from the start.
I hope this helps! If something is not clear or correct, please leave a comment.