Shakespeare is using a metaphor here wherein Romeo compares his meeting and falling in love with Juliet to a ledger, or a record of debts and payments. As my colleague stated, Romeo has just learned Juliet's identity. She is a Capulet, his families sworn enemy. 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.
Enotes has wonderful side-by-side texts for Shakespeare's plays. They have the original text and a modern translation. Check out the link below for the e-text for Romeo and Juliet.