Here is the full list of all 22 GUNHEAD achievements.
Fail a mission
Complete all bonus objectives on a ship
Defeat an enemy ship under the goal time.
Defeat The Tyrant
Defeat The Juggernaut
Defeat Gemini
Defeat The Python
Defeat The Matriarch
Defeat The Observer
Defeat The Kraken
Defeat all bosses
Unlock the Archangel
Unlock the Jennet
Unlock the Rook
Unlock the Shinobi
Unlock the Tigersnake
Unlock all suits
Get a license
Get all licenses
Finish a campaign on Medium difficulty
Finish a campaign on Hard difficulty
Finish the second loop of a campaign