Here is the full list of all 20 Explodemon achievements.
S-Rank all Squall levels
Beat the par times for all levels
S-Rank all Arbor levels
Destroy 15 enemies in less than five seconds
Completely upgrade Explodemon and unlock all of his abilities
Close Explode the same green sticky 10 times
S-Rank all Nibia levels
Complete Adventure Mode
Destroy 1000 Vortex
Destroy 100 enemies by dunking them in acid
Beat a par time for any level
Bring down a Vortex dropship
Perform a 3-Hit Combo
Complete any level with an A-Rank or higher
Destroy 10 enemies in less than five seconds
Explode 10 times in the air without touching the ground
Defeat Absorbemon without taking any damage
Destroy five enemies in less than five seconds
Level up to Max Level
Perform a Counter