There is the long way, and the short way (you also get another achievement "I Know You")
Either way, you must start the normal shift. As soon as possible, open the cupboard with the paper strips to detect sigils. You might get 1 or 2 during the first body embalment.
Once you have 2 sigils, you normally can already narrow it down to only a few possible Demons.
Continue with the story and the embalments until you get the 3rd sigil. Most likely you will have a name now.
If you try to figure out which is the demon body to burn (and are correct) then you will finish this shift successfully and get this achievement plus "I Know You". If on top of that, you were also quick, then you can even get the speed run achievement.
In the case that you want to embalm all bodies to be sure, then proceed to every body, which normally will mark exactly which body was the demon.
IMPORTANT: Dont forget to place the Mark on the body before burning it!