Here is the full list of all 25 The Wrath of the Goose King achievements.
Completed the first act of the game
Completed the second act of the game
Completed the third act of the game
Complete a run with Pandarina in your party
Complete a run with Sir Muffins in your party
Complete a run with Jawbreaker in your party
Complete a run with Pandalf in your party
Complete a run with Chiaki in your party
Complete a run with Ace in your party
Complete a run with Hellpaw in your party
Complete a run with Pawnka in your party
Complete a run with Jellybelly in your party
Complete a run with Susan in your party
Complete a run with Dr. Jynx in your party
Complete a run with Marie Curry in your party
Complete Challenge Mode 1 and watch its secret ending
Complete Challenge Mode 2 and watch its secret ending
Complete Challenge Mode 3 and watch its secret ending
Complete Challenge Mode 4 and watch its secret ending
Complete Challenge Mode 5 and watch its secret ending
Complete Challenge Mode 6 and watch its secret ending
Complete Challenge Mode 7 and watch its secret ending
Complete Challenge Mode 8 and watch its secret ending
Complete Challenge Mode 9 and watch its secret ending
Complete a run with Antoine in your party