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.
Slew the Captain Dragon.
Found Charm Stone in Mouse-Man's secret location.
Slew the Dragon Zombie.
Slew the Mummy Dragon.
Slew the Meka Dragon.
Slew the Daimyo Dragon.
Slew the Vampire Dragon.
Cleared the game (Hard mode).
Found Charm Stone in Lizard-Man's secret location.
Found Charm Stone in Piranha-Man's secret location.
Found Charm Stone in Lion-Man's secret location.
Found Charm Stone in Hawk-Man's secret location.
Found Charm Stone in Hu-Man's secret location.