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