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I posted this initially as a comment on the other solution but figured it would be much better to make a separate solution for this.
After unlocking this achievement myself, and reading all over the internet, I came to the conclusion that you only need Arthur's playthrough (act 1) for this achievement.
Now, I will explain how I found out.
In my first playthrough, I decided to just pretty much slaughter everything that gets in my way if I can't stealth past it, and I even got the 100 kills achievement before I even finished the first act.
After finishing the game (all 3 acts) I restarted the game (using the "new game" option) and played through Arthur's act without killing anyone (keeping the murder stat at 0, non-lethal "kills" are okay, and I had plenty of those).
As soon as I finished the first act (and finished Arthur's story) I unlocked the achievement. It was during the load for the second act.
According to many sources, this is a common occurrence and while it might be a glitch (and it wouldn't be a stretch because this game has a lot of bugs) I still think it is necessary to let you people know that you don't have to go through all of it without any murders. Some parts (like the beginning of act 3) will truly test your patience and make you pull your hair out trying to do it non-lethally.
Might be a glitch, but you only need to finish Arthur's playthrough without any murders.
Personally, it worked for me (and others in the comments) by finishing the game killing freely and then replaying only Arthur's act, and upon completion of the first act, unlocking the achievement.