Here is the full list of all 19 Rheum achievements.
Beat the game
Beat the game on hard difficulty
Get 50 close kills
Beat the game without taking any upgrade shields
Beat level 1
Beat level 2
Clear level 1 without taking damage
Clear level 2 without taking damage
Beat the game without taking damage
Kill 1000 enemies
Kill 5000 enemies
Kill 100000 enemies
Release an eye from Keeper of Eye
Release an eye from Lord Sphincter
Beat the game with a new character
Be in close proximity to enemy bullets for 1 minute