Here is the full list of all 22 Dog Duty achievements.
The game finished before even starting! How innovative!
Destroyed the Octo Tank
Destroyed the Octo Jumper
Destroyed the Octo Plane
Finished the game
Found a deeper layer
Cleared all locations
Rescued Rebecca
Rescued Paula
Rescued Madness
Conquered all armor depots.
Stacked a good amount of enemy goods
Destroyed all enemy communications devices.
Won a battle with every vehicle.
Defeated an enemy squadron on the Hoovercraft.
Defeated an enemy squadron on the Boat.
Defeated an enemy squadron on the Truck.
Found an abandoned car in the Desert.
Your vehicle has been armed.
Rescued Kilmer's bootleg character.
Rescued Liza.