DOOM + DOOM II

10/10/24 Update:
As of Update 1, you can now easily activate cheats on console through the pause menu.

EDIT: Thanks to all the comments down below for helping people with the original enemies and more. I've overhauled this solution to include easy locations for just about every enemy in the collection.

Cheats:
It goes without saying that using the in-game cheats makes this achievement trivial.

If playing on PC: use the `~ key to open the command line and use 'god' for godmode, 'give all' for all weapons and items, and 'noclip' to walk through walls.

If playing on Xbox consoles: plug in a keyboard via USB and while the game is running, type 'IDDQD' for godmode, 'IDKFA' for all weapons and items, and 'IDCLIP' for noclip.

Full list of original Doom & Doom II enemies needed (17 total):
Zombieman, Shotgun Guy, Imp, Demon, Spectre, Baron of Hell, Cacodemon, Lost Soul, Spiderdemon, Heavy Weapon Dude, Hell Knight, Revenant, Mancubus, Arachnotron, Pain Elemental, Arch-Vile, Cyberdemon

As HUN playmore pointed out, Doom II Map30 Icon of Sin has almost every original enemy. Set the Difficulty to Ultra-Violence and turn on Fast Enemies in the Difficulty Modifiers and load up the level. Give yourself godmode and all weapons. On Ultra-Violence, these are all the enemies that will spawn from the Icon of Sin: Imp, Demon, Spectre, Cacodemon, Mancubus, Hell Knight, Arachnotron, Baron of Hell (rare), Revenant (rare, but one is already spawned at the start on U-V difficulty), Pain Elemental (rare, but spawns Lost Souls), and the Arch-Vile (very very rarely). If you're not finding the enemies you're missing just wait it out, I confirmed all of these do spawn in eventually.

Enemies like the Zombieman, Shotgun Guy, and Heavy Weapon Dude can be found in basically any level of Doom II except Icon of Sin.

Go back to Doom 1 and load up E3M8 (Dis) to kill the Spiderdemon boss.

The easter egg enemies (Wolfenstein SS / Commander Keen) and John Romero's head aren't counted.

Legacy of Rust contains new enemies not found in the original Dooms 1 & 2. The new enemies are:
Banshee - floating screaming red blob with a face
Ghoul - based off an early Lost Soul design
Shocktrooper - looks like the player model
Mindweaver - based off an early Arachnotron design
Vassago - large black demon that shoots fire
Tyrant - green variant of Cyberdemon

For the these new enemies added in Legacy of Rust + rare original enemies, start by loading up E2M7 The Brink of that game. Give yourself godmode, all weapons, and use noclip to bypass the doors. In this level you'll find: Cyberdemon, Arch-Vile, Vassago, and Tyrant. The Cyberdemons are behind doors in the first arena areas and there's a pile of Arch-Viles behind a central door further down the line.

Next load up Legacy of Rust E1M6 The Coiled City to find Mindweavers and Ghouls at the bottom of the spiral stairs (hit the switch on the giant skull on the wall).

Legacy of Rust E1M7 Forfeited Salvation has both Shocktroopers and Banshees hidden inside the walls and behind doors, so use noclip to find them. Banshees suicide-bomb themselves when close to you so hold down shoot while entering their rooms.

If there's anything useful I should add here let me know.

(all levels are in Legacy of Rust)
There are Banshees and Ghouls right at the start of E2M5. Shocktroopers can be found in E1M0 Ash Mill behind the big windmill. Mindweavers are the first enemies you'll encounter on E1M6. E2M7 contains both Vassago and Tyrants. Use noclip to find all these enemies more quickly.

The enemy I was missing was The Banshee. Here is where to find it easily

Legacy of Rust contains new enemies not found in the original Dooms 1 & 2.

The new enemies are:
Banshee - floating screaming red blob with a face
Ghoul - based off an early Lost Soul design
Shocktrooper - looks like the player model
Mindweaver - based off an early Arachnotron design
Vassago - large black demon that shoots fire
Tyrant - green variant of Cyberdemon

(all levels are in Legacy of Rust)
There are Banshees and Ghouls right at the start of E2M5. Shocktroopers can be found in E1M0 Ash Mill behind the big windmill. Mindweavers are the first enemies you'll encounter on E1M6. E2M7 contains both Vassago and Tyrants. Use noclip to find all these enemies more quickly.

The original enemies:
Zombieman, Shotgun Guy, Imp, Demon, Spectre, Baron of Hell, Cacodemon, Lost Soul, Spiderdemon, Heavy Weapon Dude, Hell Knight, Revenant, Mancubus, Arachnotron, Pain Elemental, Arch-Vile, Cyberdemon

The easter egg enemies (Wolfenstein SS, Commander Keen, and the final boss head) aren't needed.

There are 23 enemies to kill. Apart from all the enemies you expect in DOOM and DOOM II, Legacy of Rust also bring several new enemies. You'll fight some on the last stage (E2M7); E2M5 is a handy place to get the Banshee enemy, which I was struggling to find as my last one.