All you have to do is go to Nyan Nyan ninja mode in main menu. This is the easiest mode by far because you don't have to remain in suspense as all the cards are exposed and not turned over.

To get the achievements you have to double click on every unique card to show a description of that particular cat.
Achievements will unlock accordingly.
Continue doing this for all varieties of cats and you're done!

A point of note:

It is not possible to get stuck in this game. Just try your luck in different sets and eventually you will have all your sets!

Now there is a time limit so try to be as fast as you can. But there is no harm done even if you miss the timer, you can easily try again in the next set to get the rest of achievements.

Tell me which your favourite cat is! Mine is Lady Selena as she is so classy! And she reminds me of Catwoman.