Juliet's father, Lord Capulet, actually arranges for Count Paris to meet Juliet twice. First, in Act I, he encourages Paris to see Juliet at the party Capulet is giving, and in Act III, after he has decided the two should be married he tells Paris to come to Capulet's on the day of the wedding. The two actually only meet once outside of Friar Lawrence's cell in the opening scene of Act IV.
In Act I, Scene 2, Count Paris, who is obviously wealthy and a good match for Juliet, asks Lord Capulet for the girl's hand in marriage. It really doesn't matter that she is only 13 years old. Paris says that even younger girls are already having children. It was typical of that time for older men to marry young girls, and in fact, we are told that Capulet had married his wife when she was quite young. Capulet is hesitant to give his permission and advises Paris to get to know his daughter and persuade Juliet to fall in love with him. Capulet says,
But woo her, gentle Paris, get her heart; My will to her consent is but a part.
Shakespeare never tells us whether Paris talks to Juliet at the party. Instead the party scene, Act I, Scene 5, is dominated by the meeting between Romeo and Juliet.
In Act III, after the deaths of Mercutio and Tybalt, Capulet finally gives his consent for Paris to marry his daughter. Of course, Juliet is already married. When Juliet seeks advice from Friar Lawrence in Act IV, Scene 1, she meets Paris, who is under the impression that they will soon be married. Juliet hides her true feelings and attempts to quickly end the conversation.
The last arranged meeting is interrupted by Juliet's supposed death. Paris comes to Capulet's for the wedding but the Nurse has discovered Juliet dead in her room. We realize for the first time how much Paris really loves Juliet when he says,
Beguiled, divorcèd, wrongèd, spited, slain!
Most detestable death, by thee beguiled,
By cruel, cruel thee quite overthrown!
O love! O life! Not life, but love in death!
The last time we see Paris he is confronting Romeo
at the tomb. Because he knows nothing of the secret love he thinks Romeo is vandalizing the Capulet vault. They fight and Paris becomes one of the six casualties in Shakespeare's play.