How does Romeo behave in Act 5 Scene 3 of Romeo and Juliet?

The beginning of Act 5, scene 3 of Romeo and
Juliet marks Romeo's arrival at the Capulet monument.  It is clear, from
Romeo's interactions with the other characters present in the scene, that he is firmly
intent on committing suicide to be with Juliet.  First, Romeo instructs Balthasar that
no matter what he (Balthasar) hears or sees, he must not interrupt
Romeo:

Give me
the light. Upon thy life I charge thee,(25)
Whate'er thou hearest or seest,
stand all aloof
And do not interrupt me in my
course.

Then, Paris, who is
visiting Juliet's tomb and recognizes Romeo as Tybalt's murderer, exchanges words with
Romeo. When Paris tells Romeo that he must die, Romeo, who intends to kill himself to be
with Juliet, agrees and insists that he has come to the tomb to die.  Romeo gives Paris
the opportunity to avoid a fight, and says,

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Good gentle youth, tempt not a desp'rate man.

Fly hence and leave me. Think upon these gone;(60)
Let them
affright thee. I beseech thee, youth,
Put not another sin upon my head

By urging me to fury. O, be gone!
By heaven, I love thee better
than myself,
For I come hither arm'd against myself.(65)
Stay not,
be gone. Live, and hereafter say
A madman's mercy bid thee run
away.

However, Paris, unaware
of Romeo's role in Juliet's life, refuses to leave and is killed by
Romeo.

Ultimately, the play ends with Romeo's suicide,
Juliet's awakening, and her subsequent suicide.  It is clear, from this scene, that
Romeo is steadfastly determined to take his own life in order to be with Juliet. He does
not allow Balthasar or Paris to distract him from his ultimate
goal.