There are two places in the play in which we see Paris visiting Lord Capulet privately. The first visit takes place in the second scene of Act 1. Since Paris ends his first comment to Lord Capulet with, "But now, my lord, what say you to my suit?," we know that his one and only intention for visiting Capulet that day is to ask Capulet for his daughter's hand in marriage. Not only that, we know that this is actually not the first time he has asked Capulet for permission to marry Juliet. We know this from Capule's response:

But saying o'er what I have said before:
My child is yet a stranger of the world;
She hath not seen the change of fourteen years,
Let two more summers wither in their pride,
Ere we may think her ripe to be a bride. (I.ii.7-11)

The line, "But saying o'er what I have said before," can be translated to mean "by saying again what I already told you before." In other words, Capulet is reminding Paris that he has already asked him that question and that he has still not changed his mind about the subject. Capulet believes that at the age of 12, Juliet is far too young to marry.

The second time we see Paris visit Capulet it seems that Paris has  come for two reasons. First he wants to extend his sympathy to the family for their recent loss of Tybalt. Secondly, he wants to see what impression he made on Juliet at the feast and if she is willing to accept his offer. We know that he wants to extend sympathy to the family because his first line refers to the family's grief, through the phrase "these times of woe." However, we also know that he has come again to pursue his interest in Juliet because he further says,

These times of woe afford no tune to woo.
Madam, good night. Commend me to your daughter. (III.iv.8-9)

Not only that, we also know that he has come to pursue Juliet because Capulet's first lines to him are, "Things have fall'n out, sir, so unluckily / That we have had no time to move our daughter." The word "move" in this line can be translated as "persuade." In other words, Paris is saying that due to her grief over Tybalt's death, he and Lady Capulet have not had a chance to try and encourage Juliet to accept Paris's offer.