Here is the full list of all 25 Blazing Beaks achievements.
Kill Bogy
Beat a boss without taking damage
Find a secret level
Kill Newt
Open a chest
Pick up at least 25 artifacts before looping
Kill Crux
Kill Muck
Kill Larva
Kill Evil Teal
Loop once
Loop twice
Buy three weapons during the same trip to the shop
Revive a player in co-op
Collect every single weapon
Use a coin machine at least once
Complete the game with the Egg
Beat the game without collecting any artifacts
Trade in all your HP at heart machine
Kill every enemy at least once
Reach a max. HP of 15
Collect enough artifacts to reach a risk value of 10
Get killed by a ghost
Annoy the Crow at least three times during the same trip to the shop
Steal a weapon