Here is the full list of all 13 Wonder Boy: The Dragons Trap achievements. It takes around 8-10 hours to unlock all of the achievements on Windows.
Found Charm Stone in Hu-Man's secret location.
Slew the Vampire Dragon.
Cleared the game (Hard mode).
Found Charm Stone in Lion-Man's secret location.
Found Charm Stone in Piranha-Man's secret location.
Found Charm Stone in Lizard-Man's secret location.
Found Charm Stone in Hawk-Man's secret location.
Found Charm Stone in Mouse-Man's secret location.
Slew the Daimyo Dragon.
Slew the Captain Dragon.
Slew the Dragon Zombie.
Slew the Mummy Dragon.
Slew the Meka Dragon.