Most readers understand Act III, Scene 1, as the climax of Romeo and Juliet due to the increased speed of the action in the events. After Tybalt kills Mercutio, Romeo is full of emotion and wants to have a duel with Tybalt.
And fire-eyed fury be my conduct now!
Now, Tybalt, take the 'villain' back again
That late thou gavest me; for Mercutio's soul
Is but a little way above our heads,
Staying for thine to keep him company.
Either thou or I, or both, must go with him.
While Romeo and Tybalt fight, Romeo kills Tybalt with his sword. This is the highest action point in the play;thus, it is the climax. All the events from here onward are part of the falling action of the play. When Prince Escalus inquires about the death of Tybalt and Mercutio at the end of Act III, Scene 2, he is told that Romeo is responsible for the death of Tybalt; therefore, the Prince banishes Romeo.
The climax of Romeo and Juliet occurs in Act 3 just after Romeo kills Tybalt and says "O! I am Fortune's Fool!" when he realizes he has just murdered his wife's cousin.
We know this is the climax of the play not only because it represents the greatest tension in the entire plot (as readers, we ask ourselves at this point, "How in the world is Romeo going to get out of this blunder?"), but also because every event from this point forward is resolving (or at the very least seeking to resolve) some piece of prior conflict in the play.