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.
Acquire all 3 Demon Key Artifacts in the main campaign
Finish the Lost Levels mission
Finish 'No Escape' on Watch Me Die skill level
Defeat the Mother of All Demons on The Absolution
Finish a 'Fun' Level
Locate the secret level 'Hectic'
Kill a Cyberdemon
Find the Unmaker
Find the BFG9000
Discover a secret area