First, you choose a "level" with wichever cat you want to play with. They have a certain number of stars; the fewer stars, the easier the level will be. Then you can start with the game: the cat you chose will send you a word in Japanese and you've got to answer with another that overlaps with it's last "syllables" (mora). The rules are that you can't send a word twice (that you've sent already) or a word finishing with ん (because there are no words starting with this character).
There's also a menu for setting the game as you like; you can change it's speed, if you want to play with kanji, or even the background.
By the moment, that's all I know about Shiritori Cat. I hope I could help