Here is the full list of all 20 DOOM (1993) trophies - there are 5 bronze, 6 silver, 8 gold and 1 platinum.
Unlock all trophies in DOOM
Beat every level on Nightmare in co-operative mode
Find all the secret levels
Beat all the levels in "Thy Flesh Consumed"
Beat DOOM in under 2 hours
Find a secret level
Beat all the levels in "Inferno"
Kill one of every enemy
Kill only a single enemy with a shot from the BFG
Get 100 kills in deathmatch
Get 25 kills in deathmatch
Finish a level with 200% armor and 200% health
Finish off a Cyberdemon with the pistol
Beat all the levels in "The Shores of Hell"
Kill 5 enemies in 5 seconds with the plasma rifle
Get 25 kills with your fists in a single level
Beat all the levels in "Knee-Deep in the Dead"
Crush a corpse in a door
Kill 2 enemies with a single barrel explosion
Get 3 multi-kills with the shotgun