Having up-to-date knowledge of the blockchain means each client knows which unspent transactions are available. When the next block arrives, each transaction's inputs can be compared to available outputs (unspent txns). Each unspent transaction is processed as a whole, so each is registered as spent exactly once, and a new transaction may redirect some "change output" back to the same owner if not everything is transferred.

Each bitcoin client will itself qualify each block to see if it checks out completely and onlu then is it accepted as part of the blockchain. That way, the blockchain can only be extended with valid blocks, i.e. no cheating/fraud permitted, verified by each bitcoin client. (Referring here to full clients that themselves process the blockchain.)