This should come naturally as you play. On the ultraviolence difficulty you should have come across every enemy there is by the middle of Map11, as there are no special secret blue enemies in the secret maps, and the final boss doesn’t count.
However, what may not be obvious, the game needs to register the enemies as killed by the player! So, if you do like I did and let the ceiling crushers and enemy infighting do the killing for you, the kills don’t register. For example, in Map06 (Crusher), on UV difficulty, you can let a certain enemy be killed by a ceiling crusher, and the next opportunity to kill that particular enemy comes as late as Map20, which is where I finally got the achievement.
Here’s checklist for enemies, in order of appearance (according to my memory) on the Ultraviolence difficulty.
1. Pistol zombie
2. Shotgun zombie
3. Chaingun zombie
4. Imp (brown fireball guys)
5. Demon (pink pig)
6. Spectre (invisible pink pig)
7. Lost soul (flaming skulls, although I’m not sure if these count, but you’ll surely be killing these).
8. Cacodemon (floating tomatoes)
9. Hell knights (goats who say ”aurora” when they die)
10. Revenant (skeleton guys)
11. Spider mastermind (big spiders). You’ll first meet this guy on Map06 (on ultra-violence difficulty).
12. Arachnotrons (small spiders)
13. Mancubus (fat guys who say ”cannibael-o”)
14. Baron of Hell (stronger goats)
15. Cyberdemon (the big goat with rocket launcher). You’ll first see these in Map08.
16. Pain Elemental (floating moldy potatoes who spit lost souls at you).
17. Archvile (skinny guys who use magic at you and say ”Avatar!” when they die). You’ll meet first of these guys in Map11 (UV).
Edit 05/04/2020: There has been an update on the game, which restores the exclusive blue enemies to the secret levels 31 and 32. It has been confirmed (thanks xNeo21x) that these enemies are not required for the achievement.
Additionally, in case you have missed any kills, the update also included a Level select functionality, which allows you to choose and enter any level to mop up any missed enemies. The most probable misses are the rarer monsters, Cyberdemon and Spider Mastermind (others you should come up with very often and in large numbers), which both can be found near the start of level 20 ("Gotcha!"). Remember, that they must die from your weapon fire, not from attacks by other enemies.
Edit 04/05/2020: Note that the my advice of using level select is based on the conjecture that the achievement tracker keeps on working. Some have reported that level select hasn't been helpful, and that they have had to start an actual new game.
Also, when going for achievements, make sure you have add-ons (such as Evilution, Plutonia, etc.) disabled, as the achievements are disabled with them.