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.
Free all the caged cats
Free Snicker
Collect 200 fishbones in Level 5
Collect 1000 fishbones
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
Break 10 vases
Throw 5 rats into the graveyard holes
Find Cooper's book
Hit dogs 10 times
Collect 200 fishbones in Level 2
Free Daisy
Find Oskar's ties
Jump over chihuahuas 10 times
Throw 20 apples
Collect 200 fishbones in Level 1
Find Henri's book
Free Tonks