Successfully complete one of the following tasks to get a trophy:

Untouchable (Bronze): Go unharmed for an entire level, without restarting
from a checkpoint. 
Turbo (Bronze): Complete the game in less than 80 minutes, in one session 
without restarting from a checkpoint. 
Vigilante (Bronze): Eliminate 20 bad guys in 101 seconds. 
Retaliator (Bronze): Perform 200 counter attacks. 
Exterminator (Bronze): Perform 250 finishing moves. 
Shield (Bronze): Dodge or block 25 incoming attacks. 
Psycho (Silver): Complete the game in Rorschach mode. 
Hero (Silver): Complete the game in Nite Owl mode. 
Thief (Silver): Find all abilities and pick-ups in Rorschach mode. 
Detective (Silver): Find all abilities and pick-ups in Nite Owl mode. 
Teamwork (Gold): Complete the game in co-op mode.