Here is the full list of all 31 Frogun achievements.
Buy all gallery images
Buy all hats
Get all of the complete emblems
Complete the bestiary
Beat the Brutal Rush
Beat the Boss Rush
Beat Beelzebub
Win all races
Get all health upgrades
Beat Hexy
Beat Scorchy
Defeat a boss cosplaying as them
Beat Wintry
Own the Kiwi and Melon hats
Beat Kicky
Take a photo of the scenery
Get a Max. Health Up
Beat Prickly
Buy a hat
Hit Jake with something during a race
Pose for the camera
Finish a level without damaging any enemy
Win a race
Fall on water
Finish a level with the obsidian skull
Finish a level with all of the emblem pieces
Finish a level under the time limit
Finish a level without dying
Get all coins in a level
Finish a level with both eyemeralds
Finish a level having found the notes