Romeo comes to the churchyard, because he thinks, like everyone else except the Friar, that Juliet is dead. He is bent on killing himself, while lying next to Juliet in her grave, because of his grief. He comes into the graveyard, and tells his man Balthasar to wait and to not interfere with anything he does. Romeo is determined to commit suicide, and will not allow anyone to stop him. Paris, who has come to lay flowers on Juliet's grave, sees Romeo and tries to stop him. Paris wants to arrest Romeo, because he knows that Romeo has been banished from Verona and is thus in violation of his sentence. In addition Paris has motivation to punish Romeo, because he thinks that Romeo is the cause of Juliet's "death". It was assumed by everyone that Juliet died over grief at Tybalt's death, and Romeo killed Tybalt.
So Paris comes toward Romeo, threatening to bring him to the Prince. Romeo does not take well to being waylaid in his plans, and attacks Paris. They fight, and Romeo kills Paris. Before he dies, Paris asks
PAR: O, I am slain! If thou be merciful,
Open the tomb, lay me with Juliet. (V.iii. 72-3)
Romeo does this, and recognizes the he has killed his friend Mercutio's kinsman. While looking on Juliet Romeo is amazed and how lifelike and beautiful Juliet is, still, in death. He doesn't know that she is only in a potion-induced coma. Romeo can no longer bear his grief, and drinks the poison he has brought with him.
Rom: Come, bitter conduct; come, unsavoury guide!
Thou desperate pilot, now at once run on
The dashing rocks thy seasick weary bark!
Here's to my love! Drinks. O true apothecary!
Thy drugs are quick. Thus with a kiss I die.(120) (V.iii.116-120)