There are 13 Trophies that can be collected.

BRONZE

Level Entry
Complete any Level in any difficulty in singleplayer.

100% Level
Complete a Level with 100% Kills, Items and Secrets in singleplayer.

THE BIG GUN
Find a BFG in singleplayer.

Finder of Secrets
Find a Secret in any Level in singleplayer.

Knuckle Sammich
Kill a CyberDemon with your fists in singleplayer.

DOOMed Lumberjack
Kill 20 enemies with your chainsaw on a single map in singleplayer.

Berserker
Kill 20 enemies while in berserker mode on a single map in singleplayer.

SILVER

DOOMed to Die Young
Complete the Ultimate DOOM single player campaign on "I'm Too Young to Die!" or higher.

DOOMed to Die Young Again
Complete the DOOM 2: Hell on Earth single player campaign on "I'm Too Young to Die!" or higher.

Evilution Theory
Complete the Final DOOM: TNT Evilution single player campaign on "I'm Too Young to Die!" or higher.

Plutonia is DOOMed
Complete the Final DOOM: Plutonia Experiment single player campaign on "I'm Too Young to Die!" or higher.

Master Levels Master
Complete the DOOM 2: Master Levels single player campaign on "I'm Too Young to Die!" or higher.

Dead and Rested
Complete the DOOM 2: No Rest for the Living single player campaign on "I'm Too Young to Die!" or higher.

http://www.ps3trophies.org/game/doom-classic-complete/trophies/