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.
Play the piano game with less than 3 mistakes
Collect 200 fishbones in Level 5
Collect 1000 fishbones
Find Nala's cup
Hit dogs 10 times
Jump over chihuahuas 10 times
Break 10 vases
Free all the caged cats
Free Snicker
Find Nora's score
Collect 200 fishbones in Level 4
Free Abb
Collect 200 fishbones in Level 3
Free Sheldon
Find Cooper's book
Collect 200 fishbones in Level 1
Throw 5 rats into the graveyard holes
Find Oskar's ties
Collect 200 fishbones in Level 2
Free Daisy
Throw 20 apples
Find Henri's book
Free Tonks