There are 12 Trophies that can be collected.

Bronze

FLAWLESS 2X    
Win your first double flawless victory againt CPU

FINISH HIM!    
Perform your first Fatality

GOO GOO GA GA    
Perform your first Babality    

BEWARE OF THE BEAST    
Peform your first Animality    

CAN'T WE ALL JUST GET ALONG?    
Perform your first Friendship    

PERFECT STRANGERS    
Win a flawless victory in a ranked kombat    

KONNECTED KILL    
Win your 1st online 1v1

Silver

ARCADE KOMPETITOR    
Finish Mortal Kombat    

ARCADE KOMBATANT    
Finish Mortal Kombat II    

ARCADE KONQUEROR    
Finish Ultimate Mortal Kombat 3 on any difficulty

ELDER GOD    
Win 200 matches

Gold

THE CHOSEN ONE    
Finish all three games

http://www.ps3trophies.org/game/mortal-kombat-arcade-kollection/trophies/