Here is the full list of all 23 The Purring Quest achievements. It takes around 3-4 hours to unlock all of the achievements on Windows.
Jump over chihuahuas 10 times
Hit dogs 10 times
Break 10 vases
Free all the caged cats
Collect 200 fishbones in Level 5
Collect 1000 fishbones
Free Snicker
Find Nala's cup
Play the piano game with less than 3 mistakes
Collect 200 fishbones in Level 4
Find Nora's score
Free Abb
Collect 200 fishbones in Level 3
Free Sheldon
Find Cooper's book
Collect 200 fishbones in Level 2
Find Oskar's ties
Throw 20 apples
Free Daisy
Throw 5 rats into the graveyard holes
Collect 200 fishbones in Level 1
Find Henri's book
Free Tonks