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