In order to achieve the happy sun achievement you need to click on the sun really fast in order to make it spin faster. Once it's spinning around fast enough you'll get the "happy sun!" achievement.

In order to achieve almost all of the achievements, you need to put a hat on every cats head. This should not take longer than 5 minutes to finish (but most likely less than that).

When you've put a hat on every cat, you will recieve 22 out of the 24 achievements.
This is what some of them look like:

In order to get the "Winter!" achivement you need to put the hats on all the cats.
Once you've put all the hats on all the cats, it will snow, and the game is over.