Let's back up just a bit to understand why Paris comes to be at the tomb.
Paris has arrived at the Capulet estate because he thinks he is about to be wedded to Juliet. Remember, Lord Capulet and Paris do not know that she has run away and married Romeo.
The nurse informs Paris of her death. Paris seems geuniely mournful. He cries out,
Beguil'd, divorced, wronged, spited, slain!
Most detestable Death, by thee beguil'd,
By cruel cruel thee quite overthrown!
O love! O life! not life, but love in death! (5.4.57-60)
In his agony, Paris rushes to the tomb and arrives first, with his page. He tells the page to look out for anyone coming, and eulogizes Juliet:
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.
The obsequies that I for thee will keep
Nightly shall be to strew thy grave and weep. (5.3.12-17)
Not long after Paris's words, Romeo approaches the tomb with Bathlazar, intent on breaking in. Paris is determined to stop him, but Romeo will not be stopped. The two begin fighting. Romeo stabs Paris, who dies of his wounds. It is another tragedy, and more are still to come.