This achievement is fairly straightforward apart from one very important thing:
YOU MUST SELECT THE "GIVE A MAN A FIRE" CHALLENGE IN THE CHALLENGE SECTION BEFORE YOU START THE COLD DARKNESS GAME.
Otherwise, none of your shotgun kills will count towards the achievement.
The shotgun is located here on the map, inside a train car.
Simply hold X
as prompted to pick it up. Then all you have to do is go around the map killing 20 enemies by pressing RB
to fire your special ammo. The enemies are always spawning so don't worry too much about trying to get it out of the way. I found it best to wait until I got swarmed to do this as it clears them faster and it doesn't matter as much about noise. Incidentally this is also how I managed to get the kill 3 with one shotgun blast achievement (by accident!).
One side note: Depending on what cards you chose to use, you may have to craft the special ammo before using it. My first game I had it automatically, my second game I had to craft it.
Hope that's clear enough :)