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