Here is the full list of all 17 Pain Party achievements.
The first of many
Finish one singleplayer level
Win a multiplayer match
Defeat the time king
Defeat La Cactura
Defeat DOOMBA 9000
Defeat the Trout of Terror
Defeat Mr Frostbite
Defeat Conehead, Diverter of Traffic
Defeat The Wall of Pain
Defeat Fallen Keyboard Warrior
Get the good ending
Get the bad ending
Kill the Unnamed Goose Boss
Survive 5 waves
Survive 10 waves
Survive 15 waves