Here is the full list of all 15 JACK 'N' HAT achievements.
Get 100 diamonds.
Activate a Check Point for the first time.
Defeat 5 enemies by throwing your hat.
Jump 25 times.
Get the Crystal and clear one level.
Died 5 times.
Used a Medicine.
Defeat 5 enemies by jumping on their heads.
Throw your hat 40 times.
Cause a TNT explosion chain.
Read 3 hint signs.
Warp into a level 4 times.
Defeat a Green Gunner
Break 100 crates.
Get into a fight with Echo.