Here is the full list of all 24 Bedlam achievements worth 1,000 gamerscore. It takes around 6-8 hours to unlock all of the achievements on Xbox One.
Got a Score of at least 3500 in Rescue Run
Wish Granted in Planetfire
Killed 5 opponents in the Arena without dying
Completed the Game
Complete Rescue Run
Completed Planetfire Ascension
Escaped from Age of Attrition
Defeated the horde
Wielded the Fire Sword
Brought down the Super Saucer
Cleared the Chilli Chomper Maze
Messed with the Valve
Destroyed the Armoured Car
Reached the end of the Boulevard du Burnistoun
Reached the end of the Rue de Chambard Rouge
Completed Star Fire
Completed Rage-Quit Reactor without the Reactor exploding
Destroyed all three guard towers in Death's Dark Vale
Defeated Lanzok and freed the captive marines
Encountered Bob
Secured Sigma Sector
Destroyed the Gaian Comms Jammer
Explored the Glitch in Alive1!
Accidently' Decapitated Gortoss