Well this certainly took off beyond what I expected it to...

UPDATE: A friend of mine informed me earlier today that this walkthrough had a few mistakes in it. My mistakes!
I didn't think you guys were going to enjoy Hatogate this much!

Anyway, since it's apparently been requested by folks outside of Team Hatoful, I'm editing this post to serve as a walkthrough for the achievements in Hatogate. I've spoiler-tagged things so that you can just see the name of the achievement, which should usually give you an idea of how to get it, and then a straight-up spoiler of how to get it if you just want it right then and there.
Achievements in the walkthrough are listed in the same order they appear in the game.

ACHIEVEMENTS TO UNLOCK (and how to unlock them):
-Say Your Prayers - Unlock the Normal Ending
Use Unicycle on Lizard, Move to Shrine (center of map).
-Komodo Dragonslayer - Unlock the Bad Ending
Use Sae'krom on Lizard.
-Gigaflare - Unlock the WTF Ending
Use Sae'krom on Pudding Mix.
-Coo coooo!! - Unlock the Pudding Ending
Open the Pudding Mix, then do anything other than Using the Pudding Mix on the Lizard.
-Hatodammerung - Unlock the Good Ending
Open the Pudding Mix, then Use it on the Lizard.
-I Quit! - Unlock the Lame Ending
Use Unicycle on Lizard, Move to Entrance (lower-left of map).
-Out of Body Experience - Leave yourself behind.
Leave Self, Leave Self, Leave Self.
-Lighten the Load - You probably didn't need that stuff anyway.
Use Unicycle on Lizard, Leave Pudding Mix, (Leave Paintbrush or Use Paintbrush on something; preferably not Millet), (Use Millet on Self, Peck Millet, or Leave Millet). This will leave only Key2 in your inventory.
-Multi-Killed!! - Death aplenty for you.
Do 10 things that get you the Game Over screen. Endings that get you killed won't count towards this.
-That is No True Pudding! - Discover the label mixup in your inventory.
Use Pudding Mix on Self, Peck Pudding Mix, or Use Key2 on Pudding Mix.
-It's Millet Time - Grab a healthy bite to eat.
Use Millet on Self or Peck Millet, without Using Paintbrush on Millet beforehand.
-I Feel Pretty - Give yourself a makeover!
Use Paintbrush on Self, Peck Self. Or vice-versa.
-Can I Keep It? - Aww, I guess not.
Take Lizard.
-The Obligatory "Get All Other Achievements" Achievement
...Do you really have to ask? :b

Hope this helps, everybirdie!