Here is the full list of all 30 Batman: Arkham Origins Blackgate - there are Deluxe Edition achievements.
Get a 12 hit combo
Find all detective cases and have all pickups and upgrades
Defeat Joker
Defeat Penguin
Defeat Black Mask
Defeat Bronze Tiger
Defeat Solomon Grundy
Defeat Deadshot
Catch Catwoman in Gotham City
Defeat Catwoman
Defeat Grundy using the Shock Batarang and water
Complete your first detective case
Defuse all Joker presents
Rescue all Penguin hostages
Disarm all Black Mask junction boxes
Get a 25 hit combo
Get a 50 hit combo
Get a 100 hit combo
Beatdown 10 thugs
Perform 20 stealth takedowns
Die from inhaling gas
Unlock the Zombie Suit
Complete all detective cases
Find and destroy all Joker teeth
Find and destroy all Penguin bird cages
Find and destroy all of Black Mask's masks
Obtain all Batsuits attainable through gameplay
Find all armor upgrades
Find all RUSH upgrades
Find all gauntlet upgrades