Here is the full list of all 24 Bedlam achievements.
Escape from Age of Attrition
Defeat the horde
Find the fire sword
Bring down the Super Saucer
Clear the Chilli Comper Maze
Mess with The Valve
Destroy the Armoured Car
Reach the end of the Boulevard du Burnistoun
Reach the end of the Rue de Chambard Rouge
Exit Star Fire
Complete Rage-Quit Reactor without the Reactor exploding
Destroy all three guard towers in Death's Dark Vale
Defeat Lanzok and free the captive marines
Encounter the Accounts Dept.
Secure Sigma Sector
Destroy the Gaian Comms Jammer
Complete the secondary objective
Complete the Graxian combat training and salute your commanding officer
Kill 5 opponents in the Arena without dying
Wish Granted in Planetfire
Complete Planetfire Ascension
Get a Score of at least 3500 in Rescue Run
Complete Rescue Run
Complete the Game