Here is the full list of all 20 Cats Organized Neatly achievements. It takes around 2-3 hours to unlock all of the achievements on Windows.
Solve Chapter 4
Spin a total of 1000 cats
Click on a total of 1000 cats
Let the Whincy Cat "meow" 10 times in a row
Turn on and off the sound in the options menu 10 times in a row
Unlock 20 cats
Solve Chapter 3
Solve Chapter 2
Unlock 13 cats
Solve Chapter 1
Unlock 7 cats
Spin a cat 4 times in a row
Unlock 3 cats
Solve Chapter 5
Solve Chapter 6
Solve Chapter 7
Solve Chapter 8
Spend at least 2 hours in the game
Unlock 28 cats
Unlock 30 cats