Unlike most achievements in this game, you can get this achievement in your first run-through.
You'll need to kill a total of 26 demons. They're the flying black machines that drop bombs, and they make up the majority of enemies throughout the game. The good news is that you'll encounter just about all of them naturally through the course of the game, so there's no real need to keep a count as you go. Keep in mind though that they'll occasionally appear in locations they weren't in before. This is triggered by:
1.) Talking to the lift ticket salesman at Pissward Falls -- He'll needs 300 frozen oil to sell you a lift ticket, so you'll have to backtrack to the beginning of the game to get that much oil. Fortunately you'll most likely run into all of these new spawns when doing this.
2.) Changing the time on the big clock in Red Wash to 2:30a -- Affects Red Wash only. Since you may not have to backtrack at this point in the game, be sure to check each Red Wash location now to get every demon that spawned.
The "Point of No Return" is Pinstripe's house (and specifically entering his death date into the bed, causing it to rise). If you haven't unlocked this achievement by then, just backtrack through the game until you find what you missed, they should all be spawned in at this point. If you take the bed up to the top floor of the house, you'll have to run through the game again unfortunately.
NOTE: the demon at the top of Hangley Pass (by the house) respawns as his bomb is used in a puzzle on this location. I don't know if killing this one repeatedly counts for more than one kill -- let me know if you think this is the case. I personally had this achievement unlock one demon early and I'm wondering if this is why...