Here is the full list of all 13 Gun Chain achievements.
Press the "Play" button.
Defeat Alpha
Shoot a fireball at Alpha during his laser attack
Defeat TNK
Defeat the Demon Lord
Defeat Demon Lord without shooting an incorrect color at his sword
Defeated Gec'ler
Defeat Gec'ler without shooting a wrong color at her babies
Defeat the Dawn Keeper
Zap the Dawn Keeper by color matching two floating pillars
Defeat the Final Boss
Match (5) or more balls at once
Destroy the last ball of a level with a fireball