Here is the full list of all 20 ELYSION achievements.
Complete all stages on NORMAL
Complete all stages on EASY
Complete all stages on EASY without dying
Defeat the STAGE 6 boss "Iris"
Defeat the STAGE 5 boss "Mary"
Defeat the STAGE 4 boss "Elys"
Defeat the STAGE 3 boss "Lycoris"
Defeat the STAGE 2 boss "Hudie"
Complete STAGE 1
Complete STAGE 1 by defeating all enemies
Defeat the STAGE 1 boss "Ruri"
Use a Bomb
Complete all stages on HARD
Complete all stages on NORMAL without dying
Complete all stages on HARD without dying
Complete EASY mode with a combined rank of S
Complete NORMAL mode with a combined rank of S
Complete HARD mode with a combined rank of S
Complete STAGE 1 without defeating any enemies
Defeat a total of 10,000 Guardians