Romeo kills himself because he believes Juliet is dead. She only appeared dead, due to the friar's potion, but the letter that would have explained this to Romeo never reached Mantua (Act V, scenes 1 and 2).
The friar's plan was that Juliet would drink the potion, appear dead for forty-two hours, and be taken to the family tomb, while a letter warned Romeo and he came to fetch her (Act IV, scene 1). Juliet drinks and is taken for dead (Act IV, scenes 3 and 5). In Mantua, Romeo asks Balthasar whether he brings letters from the friar, and he says no; he reports only that Juliet's body sleeps in the Capulet monument. Romeo replies, "Then I defy you, stars!" orders horses hired, and pays forty ducats to an apothecary, despite Mantua's law that punishes such a sale with death (Act V, scene 1).
The reason there is no letter appears in scene 2: Friar John, who was to carry it, was locked in a house suspected of plague, could not reach Mantua or find anyone to carry it, and returned the letter to the friar. Laurence sees the danger and goes alone to the tomb.
In scene 3, Romeo breaks into the tomb, kills Paris, who tried to stop him, sees Juliet, and drinks the poison. Laurence arrives afterward; Juliet wakes, and on finding her husband dead, kills herself with his dagger.