Here is the full list of all 20 ZOMBI achievements.
Kill your first Zombie with a Molotov Cocktail
First Kill with the Crossbow
Get the XL BOB
Kill 100 zombies
Get into Buckingham Palace
Kill 50 zombies
Reach a Survivor Score of 2000 points
First Headshot with a gun
Get into the Supermarket
Pick up a backpack (Bug-Out-Bag)
Make it to the Safehouse
Kill 200 zombies
Get into the Tower of London
Get into the Nursery
Survive the Ambush at the Church
Get the XXL BOB
Hack all of the CCTV junction boxes in London
Finish Game in Chicken Mode
Finish Game in Standard Mode
Finish Game in Survival Mode