There are 40 Trophies that can be collected.

Bronze

Papa's Palace of Pain    
Complete 'Papa's Palace of Pain' in Story Mode     

Naked Terror    
Complete 'Naked Terror' in Story Mode     

Ballistic Trauma    
Complete 'Ballistic Trauma' in Story Mode     

Carny    
Complete 'Carny' in Story Mode     

Creeping Flesh    
Complete 'Creeping Flesh' in Story Mode     

Scream Train    
Complete 'Scream Train' in Story Mode     

The Fetid Waters    
Complete 'The Fetid Waters' in Story Mode     

Jailhouse Judgment    
Complete 'Jailhouse Judgment' in Story Mode     

Overkill    
Complete all levels in Story Mode     

Goregasm!    
Achieve a Goregasm     

Weak At The Knees    
Maintain a Goregasm for 2 minutes     

De-Generation    
Collect 10 LPs     

Decom-Poster Child    
Collect 15 posters     

Dis-Figurine Collection    
Collect 15 figurines     

Graphic Horror    
Collect 17 comic book pages     

Boom Headshot!    
Decapitate 200 mutants    

Toasty!    
Kill 5 or more mutants with one grenade     

Yakuza    
Kill 100 mutants 'Gangster Style' (holding gun sideways)     

Suffer Like G Did    
Complete any level with only one health segment remaining     

Hardcore    
Complete any level in Director's Cut without using any continues     

Merchant of Menace    
Buy a new weapon     

Fully Loaded    
Fully upgrade any weapon     

Power of the Mind    
Complete any level with the targeting crosshair off     

I'll Try Anything Once    
Play each minigame through to completion once

Silver

Lights, Camera, Headshot!    
Complete all levels in Director's Cut     

Caesar's Salad    
Complete all CHALLENGES for 'Papa's Palace of Pain'    

Private Dance    
Complete all CHALLENGES for 'Naked Terror'    

Doctors and Nurses    
Complete all CHALLENGES for 'Ballistic Trauma'    

Clown Shoes    
Complete all CHALLENGES for 'Carny'    

Meat Packer    
Complete all CHALLENGES for 'Creeping Flesh'    

Bad Conductor    
Complete all CHALLENGES for 'Scream Train'    

Swamp Monster    
Complete all CHALLENGES for 'The Fetid Waters'    

Jail Bird    
Complete all CHALLENGES for 'Jailhouse Judgment'    

Lab Rat    
Complete all CHALLENGES for 'Overkill'    

Never Runs Dry    
Complete a level without triggering an enforced reload (i.e. reaching zero bullets)

Gold

Be My Valentine, Jill    
Unlock all unlockables and purchase all weapons   

Perfect Victory    
Achieve an S rank for all levels in Story Mode or Director's Cut     

Do You Feel Lucky?    
Complete all levels in Classic Mode without using any continues     

Extreme Hardcore    
Complete all Director's Cut levels in Hardcore Mode

Platinum

The House of the Dead: Overlord    
Unlock all trophies

http://www.ps3trophies.org/game/house-of-the-dead-overkill-extended-cut/trophies/