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