Here is the full list of all 25 Crystal Cosmos achievements.
Finish Level Five
Get an S rank in Level Five
Finish Level Four
Finish Level Three
Get an S rank in Level Two
Finish Level Two
Get a 200 kill streak
Get a 150 kill streak
Get a 100 kill streak
Finish Level One
Get an S rank in Level One
Get a 50 kill streak
Finish Level Six
Finish Level Seven
Finish Level Eight
Get an S rank in Level Three
Get an S rank in Level Four
Get an S rank in Level Six
Get an S rank in Level Seven
Get an S rank in Level Eight
Get Obsidious Ending
Get Alabastar Ending
Buy every upgrade
Finished Level Nine
Get an S rank in Level Nine