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