Paris' challenge to Romeo probably happened for several reasons:
1. He believes that Romeo is there for a reason other than Juliet; he thinks Romeo intends to cause havoc among the remains of the dead Capulets in the tomb:
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?
Paris is willing to avenge the death of Tybalt because that is the only connection he believes Romeo has to the tomb at this point.
2. Romeo is trespassing on Capulet property. Paris had every right to want to defend his new family's property.
3. He is there to grieve his fiance's death and intends to do so privately.
Romeo does not even understand who Paris is until after he kills Paris. Unfortunately for Paris, Romeo comes madly after him.