Here is the full list of all 31 Frogun achievements.
Own the Kiwi and Melon hats
Beat Beelzebub
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
Defeat a boss cosplaying as them
Win all races
Beat Hexy
Beat Scorchy
Pose for the camera
Beat Wintry
Beat Kicky
Get a Max. Health Up
Beat Prickly
Hit Jake with something during a race
Buy a hat
Win a race
Finish a level without damaging any enemy
Fall on water
Finish a level with all of the emblem pieces
Get all coins in a level
Finish a level under the time limit
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