Here is the full list of all 16 Rocket Knight achievements.
Completed 3.2 without taking damage
Defeat 3 enemies with a single reflected projectile
Defeat 5 enemies in a single Rocket Burst or Drill attack
Completed Gold Sparkster Mode
Completed 2.1 having collected all pickups
Completed Arcade Mode on Hard
Completed Arcade Mode on Hard within 60 minutes
Completed all stages below par time
Completed 1.2 having defeated all enemies
Completed all stages
Completed Arcade Mode on Normal
Completed Arcade Mode scoring 1,000,000 or more
Complete World 4
Complete World 3
Complete World 2
Complete World 1