Paris was betrothed to Juliet, and fully expected (and wanted) to marry her. He is taking the news of her "death" very hard, and he goes to the tomb to place flowers on her grave. He is weeping and telling Juliet that every night he will come to her tomb and leave flowers, as he might have brought her flowers in the evening if she had lived and they had been married.
Sweet flower, with flowers thy bridal bed I strew
(O woe! thy canopy is dust and stones)
Which with sweet water nightly I will dew;
Or, wanting that, with tears distill'd by moans.(15)
The obsequies that I for thee will keep
Nightly shall be to strew thy grave and weep.
He adds several metaphors here, such as the canopy (over a bed) for her is dust and stones now, rather than silk and flowers over her marriage bed.
Paris sees Romeo, and, all unknowing of Juliet and Romeo's secret marriage and the element of self-defense in Romeo's slaying of Tybalt, recognizes him as a criminal banished from Verona. Paris is an upstanding citizen, and he plans to arrest Romeo and surrender him to the authorities, since Romeo is in violation of his sentence by being present in Verona. In addition to this Paris believed the Juliet died because of the death of her cousin, Tybalt. Since Paris knows nothing of the love between Juliet and Romeo, he knows only that Romeo is a Montague, the sworn enemy of Juliet's family the Capulets. Even though Juliet is dead, he is still loyal to what would have been his family-in-law, and he also feels that Romeo indirectly caused the death of Juliet by killing Tybalt. This is more than enough reason, in Paris' mind, to attack Romeo.
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.
They fight, and Romeo kills Paris. As he is dying, the lovestruck Paris begs to be laid in the tomb next to Juliet, his would-be bride.
O, I am slain! If thou be merciful,
Open the tomb, lay me with Juliet.
The emotions in this scene, as in many scenes in this play, are extremely high. Romeo, believing Juliet dead (because he never got the Friar's letter) is bent on killing himself. Paris is sorrowing deeply because he believes his fiance died on the eve of their wedding, cruelly struck down because of a consequence of a feud between her family and the Montagues. The atmosphere in the churchyard was a tinderbox, and it only took a little spark for things to explode and more deaths to be caused.