Here is the full list of all 14 Barry the Bunny achievements.
Get 25 carrots.
Get 50 carrots.
Get 100 carrots.
Reach 5 bunnies at home.
Reach 10 bunnies at home.
Reach 25 bunnies at home.
Reach 50 bunnies at home.
Defeat the first boss.
Defeat the second boss.
Defeat the third boss.
Defeat the fourth boss.
Defeat 25 enemies.
Defeat 50 enemies.
Defeat 100 enemies.