Romeo learns that Juliet is a Capulet at the conclusion of Act I, Scene 5.
Interestingly, Romeo senses fate at the end of Scene 4 of the first act. As Benvolio, Mercutio, and Romeo make their way to the Capulets, Benvolio worries that they will be too late, but Romeo has a sense of something else:
I fear too early. for my mind misgives
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date. (1.4.107-109)
Romeo has a foreboding that this night is the beginning of something that will end with his death: "...expire the term/Of a despised life...." (1.4.110-111).
Nevertheless, he decides to go along with Mercutio and Benvolio. Not long after they arrive at the party, Tybalt recognizes Romeo as a Montague, and he wishes to run his sword through Romeo. Fortunately, Lord Capulet prevents Tybalt's rash action.
In the meantime Romeo has approached Juliet and stolen a kiss from her. After this brief encounter, the Nurse approaches Juliet and tells her that her mother wishes to have a word with her. Juliet then departs and Romeo asks the Nurse who Juliet's mother is. The nurse replies in the beginning of a loquacious answer that the mother is the lady of the house. Taken aback, Romeo asks,
Is she a Capulet?
Oh, dear account! My life is my foe's debt. (1.5.117-118)
Romeo immediately has the premonition that his life is in the hands of his enemy.