Here is the full list of all 13 HellStar Squadron achievements.
Unlock Niji.
Attain a x50 combo.
Destroy a total of 500 enemy ships.
Destroy a total of 250 enemy ships.
Clear a stage without getting hit.
Attain a x100 combo.
Beat the game with Azure.
Beat the game with Citrine.
Beat the game with Emerald.
Beat the Game with Niji.
Beat the game on Hard.
Beat the game on Veteran.
Beat the Boss Rush mode.