As an intellectual exercise making a browser based checkers game seems like a fun idea. However I have my doubts as why anyone would give you money to make one. Especially one that does not enforce the rules of the game. I know a little about php and don't think is is a very good language for making some thing complex like checkers.
Flash's action script, which I am just starting to learn, seems more robust especially since it can listen for mouse events without having to swap screens. It also has a interval call back so you can have the program query the database every few seconds.
As for coding concepts for a multilayer program:
Since the internet is a vast place you will probably need some sort of match making service to connect two players and then make sure nobody else accidentally butts in on their game. Microsoft's internet gaming zone did this by having a lobby with 200 game tables. Each table had three states empty, one person waiting for a game, and two people in a game. This lobby could be represented by in a database table containing the following fields:
player1Name // some text so friends could identify each other.
player1UID // unique ID randomly generated at game start and stored in
// player's browser. The game will only accept moves from a player
// if the plaer UID matches the UID of the board.
player2Name // same description as player1
isPlayer1turn // boolean value where if false indcates player2's turn
boardState // could be a string where
// ' ' indicates an empty space,
// 'w' indicates unkinged white pieces,
// 'W' indicates kinged white pieces,
// 'b' indicates unkinged black pieces,
// 'B' indicates kinged black pieces,
If you insist on doing this in php you probably also need to remember which piece the user selected in a separate field so you know which piece to move on the user's second click. However Actions script can remember this for you so the database only needs to remember stuff that should be communicated to the other user.