Those Who Remain

To get the Executioner Achievement you need to use the matches on the gasoline on the ground beside the guy in the Police Car. To find the matches, from where the police car is, head into the building and go straight to the far wall straight ahead of the entrance, there are lockers on your left and the keys and matches can be found in there.

The achievement will pop once you go through the door to exit the area.

This video will show how to get through to the end and includes full walkthrough for all achievements.

Executioner Achievement 32:20

You get the opportunity to earn this achievement towards the end of the police station level. The escaped convict will be in the police cruiser and he gives you a task to locate the keys to help flee the scene. The keys/matches will be located in a locker in the front of the station. When reentering the station a set of lockers will swing open, the locker you are looking for is just passed those on the left side near the fire. Collect both items and head back to convict
Here you will have to make the choice of saving (give him keys) or killing (go to gas and light with match).
For this particular achievement, set him on fire, and exit through door.