Here is the full list of all 47 Batman: Arkham Asylum Game of the Year Edition achievements.
Complete one combat challenge without taking damage
Defeat Titan henchman without using Batarangs (any play mode)
Complete one predator challenge by using only Silent Takedowns and without being detected
String up one henchman and drop him to surprise a second (any play mode)
Achieve 24 medals on predator
100% Complete
Achieve 24 medals on combat challenges
Achieve 16 medals on predator challenges
Perform a perfect combo including all of Batman's combat moves (any play mode)
Achieve 8 medals on predator challenges
Catch a Batarang (any play mode)
Solve every riddle on the island
Spirit of Amadeus Arkham revealed
Solve 85% of Riddler challenges
Glide continuously for over 100m
Achieve 16 medals on combat challenges
Deliver an explosive knock out blow to the Joker (Easy difficulty)
Deliver an explosive knock out blow to the Joker (Normal difficulty)
Deliver an explosive knock out blow to the Joker (Hard difficulty)
KO all the henchmen celebrating your arrival at the party
Solve 70% of Riddler challenges
Achieve 8 medals on combat challenges
Defeat the giant Titan Ivy plant
Ride a Titan henchman and knock down 5 thugs (any play mode)
Solve 55% of Riddler challenges
Venture into Killer Croc's lair and come out alive
Conquer the effects of the Scarecrow's fear gas
Defeat two Titan Henchmen at once
Solve 40% of Riddler challenges
Capture and lock up Harley Quinn
Save Dr. Young from being killed by Victor Zsasz
Face your biggest fears and keep your sanity
Solve 25% of Riddler challenges
Gain access to Administration after it is locked down by the Joker
Defeat Bane
Survive the nightmare of the Scarecrow's fear gas
Save all the doctors in medical
Solve 10% of Riddler challenges
Complete a combo of 40 moves (any play mode)
Solve 5% of Riddler challenges
Escape from Intensive Treatment to the island surface
Survive the onslaught from the deformed Joker henchman
Rescue the guards and henchman from the Joker toxin in Decontamination
Take down Zsasz in the Patient Pacification Chamber
Complete a combo of 20 moves (any play mode)
Complete a combo of 10 moves (any play mode)
Complete a combo of 5 moves (any play mode)