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