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