The goal of Kanji Chain is to form the longest chain you can of words connected by kanji. A valid play is any kanji compound that shares at least one kanji with the preceding word, and introduces at least one new kanji to the chain. Once a kanji is shared between two words, it cannot be the link for the next word.
On beginner mode, the computer will use mostly kanji from low levels of the jōyō list, whereas on normal it will use the entire list and on higher levels it may also use non-jōyō kanji. Also, on Difficult and above, you can't use the same kanji twice in one game.