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