How and why does Romeo kill Paris?
Romeo kills Paris in the very last scene of the play, Act V, Scene iii. Paris has come to the churchyard to put flowers on Juliet's grave. Paris doesn't know (for no one does, except Friar Lawrence) that Juliet is really in a potion-induced coma. He sees Romeo, and remembers that Romeo is a murderer who has been banished from Verona.
PAR: This is that banish'd haughty Montague
That murdered my love's cousin—with which grief(50)
It is supposed the fair creature died—
And here is come to do some villanous shame
To the dead bodies. I will apprehend him.
Stop thy unhallowed toil, vile Montague!
Can vengeance be pursu'd further than death?(55)
Condemned villain, I do apprehend thee.
Obey, and go with me; for thou must die.
Paris, being an upstanding citizen, resolves to arrest Romeo and bring him to the authorities. He is also motivated by the fact that he thinks that Romeo is the indirect cause of Juliet's death. Romeo killed Tybalt (partially in revenge for Mercutio, and partially in self-defense), and since Paris thinks that Juliet died of grief over the death of her cousin Tybalt, Paris is understandably upset. He attacks Romeo.
Romeo is bent on killing himself by poison while lying next to the "dead" Juliet in her grave. He is in a dangerous frame of mind and will not be stopped. He sees Paris as only an obstacle to his goal, and when he is attacked by him he defends himself. Romeo ends up killing Paris.
PAR: I do defy thy conjuration
And apprehend thee for a felon here.
ROM: Wilt thou provoke me? Then have at thee, boy!(70)
Exit. Paris falls.
PAGE: O Lord, they fight! I will go call the watch.
Romeo has no real reason to kill Paris, but he is blind with grief and will not be set upon by Paris and dragged away to prison. Even if Romeo had tried to reason with Paris, and tell him to get out of his way and so that he may commit suicide, there is no way that a good Christian like Paris would allow another person to kill himself (this was a crime at the time.) Again, like Tybalt's death, this is not a premeditated crime by Romeo; but he causes yet another man's death just the same.