To get all the achievements in one play-through, you have to play on the Watch Me Die skill level. Alternatively, play the game on any skill level and later you can skip to the level No Escape (map 24) using the Features menu and finish it to get the achievement Escape Artist.

There are:

• 29 story maps (including 4 secret levels)
  
• 6 Lost Level maps
  
• 3 "fun" maps

Through this menu you can enable cheat options and skip to levels. Skipping to levels doesn't disable Steam achievements, but enabling cheat options does (or so it warns; I didn't test this).

To activate the Features menu:

• Enter a level password in the main menu at least once. However, I think the menu will only be temporary if you do this, meaning it will disappear if you exit the game. (Perhaps someone could confirm this and leave a comment.)
  
• Finish the first secret level (map 32: "Hectic") on any skill level by exit through the teleporter that needs all three keys to be accessed. This will unlock the menu permanently.

Once activated, access it while in a level by pressing escape and selecting it from the menu options.

Simplest Way to Finish Hectic

Because of a technical issue in the game, if you grab the soulsphere it will not spawn the monsters in the level. You can't grab it directly though - you have move toward it slowly so that you pick it up without moving on its platform, which will lower and trap you. (You can do the same with the green armour.) Once you've got it, just grab the three keys and exit the level.

Locate the secret level 'Hectic'



This involves blowing up all the barrels in the first level in order to open a secret teleport.

Check out this guide for details: Fandom - Doom Wiki: MAP32: Hectic (Doom 64)[doom.fandom.com] (see Secrets, point 4).

Discover a secret area



There are plenty in the first level and you'll have to find some of them in order to get to the secret level called "Hectic" (see above).

The easiest to find in the first level is one with a green armour:

(1) From where you start, go straight into the passage, turn left at the bend and open the first door on the left.
(2) Go up the stairs on the left as you open the door, along the long platform and on to the short one.
(3) Press "open" on the metal grid and step into the room to grab the green armour and get the secret.

Acquire all 3 Demon Key Artifacts in the main campaign



You have to play the three secret levels to get these artifacts. A big benefit of getting them is they improve the Unmaker and make it much more powerful. The secret maps can be found on these levels:

• MAP04: Holding Area
  
• MAP12: Altar Of Pain
  
• MAP18: Spawned Fear

Check out these guides for details:

• Secret levels: Outpost Omega, The Lair, In the Void + Demon Artifact locations by that one kid
  
• Fandom - Doom Wiki: Doom 64[doom.fandom.com]
  
• The Doom Wiki: Doom 64[doomwiki.org]

Find the BFG9000



Found on map 10 ("The Bleeding").

Find the Unmaker



Found on map 12 ("Altar Of Pain").

Kill a Cyberdemon



The first one you meet is on map 17 ("Watch Your Step"), but you can also find some in the fun levels.

Defeat the Mother of All Demons on The Absolution


Just play through to the end. The Mother is very easy if you pick up the invulnerability sphere and use the Unmaker with all three demon artifacts.

Finish 'No Escape' on Watch Me Die skill level



If you start and finish the game on the Watch Me Die skill level, you'll get this. Otherwise, I think it's reasonable to use the level password or Features menu to skip to this level (replaying the entire game just for this level will be tedious).

Finish the Lost Levels mission




Finish the main story to be able to access the Lost Levels under New Game in the main menu, or use the Features menu to access them for the first time and after that you'll find them under New Game.

You only need to complete the map Final Judgement to get the achievement.

(Thanks to CakeWut for the info on finishing the main story.)
(Thanks to NoodlePunk for mentioning that you just need to finish Final Judgement.)

Finish a 'Fun' Level



There are four levels (the fourth, Panic, is right at the end of the list, after the Lost Levels). Access them from the Features menu. "Fun" is an ironic description.