Here is the full list of all 10 DOOM 64 achievements. It takes around 1-2 hours to unlock all of the achievements on Windows.
Finish 'No Escape' on Watch Me Die skill level
Defeat the Mother of All Demons on The Absolution
Finish the Lost Levels mission
Finish a 'Fun' Level
Acquire all 3 Demon Key Artifacts in the main campaign
Kill a Cyberdemon
Find the Unmaker
Find the BFG9000
Locate the secret level 'Hectic'
Discover a secret area