Here is the full list of all 19 Curse of the Crescent Isle DX achievements. It takes around 3-4 hours to unlock all of the achievements on Windows.
Find one of the special messages in the password screen
Complete Speed Run Mode
Collect all 27 coins in the game
Take damage by staying underwater too long
Complete a stage in co-op
Complete the game with no coins
Complete the game as the Queen
Complete the Boss Rush Mode
Complete the game as the King
Defeat Aranomanom
Hit the third boss three times on its way up
Break one of those big rocks
Let a Carfee fall in the water
Talk to the statue in the town
Defeat Naja Naja
Defeat Grondor of the Forgotten Mine
Jump off an enemy while grabbing him
Complete a stage without taking damage
"Accidentally" kill an enemy