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