In Shakespeare, Act 3, Scene 5, can anybody find some rhyming, and analyze why it has been used?

I am assuming we are talking about end-rhyming.  In at least one spot in this scene, in which Romeo and Juliet have spent their first married night together and are joking, in verse, together about how the rising sun is really the moon, Shakespeare uses rhyming for humor and emphasis.  They are doing this because Romeo has to leave as soon as it is light, or he will undoubtedly be hunted down and killed by the Capulets in retribution for Romeo having killed Tybalt.  So the young married couple have only a few minutes together, and they are joking about how the celestial bodies of the sun and moon should cooperate and extend their time together. And though Romeo knows he must leave, he jokes that he will stay in Verona and die for Juliet's sake (something she would never want or allow).  Here, the rhyming is funny, because Romeo is saying something both romantic and improbable:

I have more care to stay than will to go.
Come, death, and welcome! Juliet wills it so. (23-24)

To make this into a rhyming couplet underlies the hyperbole of the statement, and makes the reader (and, Romeo hopes, Juliet) smile.

Juliet follows up with another rhyming couplet, in the same position in her speech (the third- and second-to-last lines)

Since arm from arm that voice doth us affray,
Hunting thee hence with hunt's-up to the day!

Juliet is continuing the joke, implying that she wishes that the lark and the toad had changed voices, and the morning lark would not bring the day to them.  The rhyming here serves the same purpose.

Juliet finishes speaking

O, now be gone! More light and light it grows.(35)

And Romeo finishes with another rhyme, showing that their time alone together is coming to an end.

More light and light—more dark and dark our woes!

As a general rule (although not 100% of the time) the higher-class characters speak in verse, and often their speeches of blank verse end in rhyming couplets.  Romeo and Juliet are of this class of people, so you are more likely to find them speaking in rhyme than some of the lower-class characters (such as the Nurse).

Later in this scene Juliet uses end-rhyming to emphasize the irony when she is talking to her mother about marrying Paris.  The audience knows that Juliet cannot marry Paris, but Lady Capulet does not.

Now by Saint Peter's Church, and Peter too,
He shall not make me there a joyful bride!(120)
I wonder at this haste, that I must wed
Ere he that should be husband comes to woo.

There's not a lot of rhyming in this scene, but when Shakespeare uses it he uses it to good effect, and usually for humor or irony.  This is true of most of the play.  As a general rule (again, this is not 100%) there is less rhyming in Shakespeare's tragic plays than in his comedies.  Romeo and Juliet is a tragedy.