This achievement has to be unlocked on your first playthrough. Pinstripe lets you delete your save file in the Options menu. If it still doesn't unlock (like it didn't for me), try deleting your save through the Manage Game menu on the Xbox. Be sure to choose the "Delete from Everywhere" option.
The video above shows my full, no-damage playthrough. Combat is pretty simple and you usually only fight one demon at a time. The demons will drop bombs after a few seconds but they're easy to dodge and will disappear when the demon dies.
These were the few areas that really gave me trouble:
The First Mini Boss (17:46)
Jumping the Gap in the Cave (24:37) / (30:57) / (40:43)
The Second Mini Boss (25:11)
The Final Boss (57:56)
I highly recommend unlocking the Infinite Songbird Rifle as it makes the final boss much easier.
This achievement can be pretty annoying so I hope this helps. If anything is missing or needs changing, please let me know.