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