Here is the full list of all 12 Rocket Knight trophies - there are 7 bronze, 4 silver and 1 gold.
Completed Arcade Mode on Hard within 60 minutes
Defeat 3 enemies with a single reflected projectile
Completed Gold Sparkster Mode
Completed Arcade Mode scoring One Million Points or more
Completed Arcade Mode on Hard in a Single Playthrough
Completed all stages below par time
Defeat 5 enemies in a single Rocket Burst or Drill attack
Completed 2.1 having collected all pickups
Completed 3.2 without taking damage
Completed Arcade Mode on Normal
Completed all stages
Completed 1.2 having defeated all enemies