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