In this example Shakespeare is using a metaphor wherein Romeo compares his meeting and falling in love with Juliet to a debt that he needs to repay. Romeo has just learned Juliet's family name. She is a Capulet. So, when he says, "Oh dear account! My life is my foe's debt," what he means is that he is forever in debt to his enemy for bringing him his love, Juliet. The account is dear in two ways. Juliet is dear to him. Also, the debt is dear, meaning very costly; he fears his enemy Capulet will not accept his "payment" of love and gratitude--that Capulet will keep him from his dear Juliet.)
