The death of Paris occurs in Act V scene 3 of this great tragedy. In this scene, we see a grief-stricken Paris at Juliet's tomb, pledging to attend his dead fiancee's tomb every night and "dew" it with his tears every night. However, the boy he has with him whistles to indicate that somebody is coming, and so he retires before Romeo enters. When Paris sees Romeo opening the tomb, he re-enters the stage to apprehend Romeo and get his revenge on him for killing Tybalt and making Juliet suffer so badly with grief. Note how Paris addresses Romeo:
Stop thy unhallowed toil, vile Montague!
Can vengeance be pursued further than death?
Condemned villain, I do apprehend thee.
Obey, and go with me, for thou must die.
Unfortuanately, Paris hasn't counted on the fact that Romeo has come here precisely so that he can die, and the insistence of Paris that he must apprehend Romeo leads to the two of them fighting, even though Romeo tells Paris that he loves Paris better than he does himself:
By heaven, I love thee better than myself,
For I come hither armed against myself.
Stay not, be gone. Live, and hereafter say
A madman's mercy bid thee run away.
In spite of Romeo's words, Paris insists on trying to capture Romeo and prevent him from going into the tomb and doing whatever he was planning on doing, and so Romeo is forced to kill Paris so that he can enter the tomb of his beloved and die with her there.