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