It entirely depends what you mean by 'climax'. The classical notion of a climax, usually thought to occur at the centre of a play (and usually a major event which entirely changes the course of events) is usually unhelpful in studying Shakespeare, because he refuses to play by generic rules!
Yet it does apply, in a sense, to Romeo and Juliet. Tybalt arrives to try and kill Romeo, but Mercutio steps in, and is hurt under Romeo's arm - at first, he seems to think it is only 'a scratch', but the wound turns out to be fatal. Romeo then murders Tybalt when Tybalt returns.
You might of course mean 'climax' as 'ending', or 'end-point', in which case the fairly obvious answer is that Romeo and Juliet both end up dead (as the prologue, which points to them as 'death-marked' lovers, promised at the very start of the play) - Romeo swallows the poison (which he bought from the apothecary) and Juliet, waking to find his body, tries to kiss his lips to poison herself, and then - that failing - stabs herself.
Yet they are not the only victims of the final act: Romeo, of course, kills Paris outside the tomb. Thus Shakespeare shows a generation of young people wiped out: Romeo, Juliet, Mercutio, Tybalt, Paris (and the bad quarto of the play even suggests that Benvolio has died!). It is, we might think, a realisation of the plague which Mercutio calls for as he makes his final exit.
The climax in all Shakespearean tragedy always occurs in Act 3. Climax in this case can be defined as the point at which the initial conflict is resolved and the falling action (the action that occurs as a result of the initial conflict being resolved) begins. Up until this point, the two families, The Montagues and the Capulets have been feuding, and riots, started mostly by the younger generation, have been breaking out.
By the end of Act 3 this is no longer the important issue. Friar Lawrence has married Romeo and Juliet in an attempt to bring the families together, but to no avail. The primary instigators of the trouble die anyway.
Tybalt comes looking for Romeo, and when Romeo refuses to fight because he is now part of Tybalt's family, Mercutio fights to defend Romeo's honor.
Romeo, in an effort to stop the fight, steps between Tybalt and Mercutio, and Mercutio is accidentally stabbed and killed.
In a rage, Romeo pursues Tybalt and kills him, and Romeo is banished.
So now there is a new crop of problems, but the initial problem is resolved.
(Just to be clear--the two dead people are TYBALT and MERCUTIO.)
For more help with Romeo and Juliet, see the links below:
Tybalt and Mercutio. Act 3, Scene 1.
While some may debate over the true climax of Romeo & Juliet, my understanding is that the true climax occurs in the last act, Act V, when Romeo rushes to the side of what he believes is his now-deceased bride.
We the readers know that Juliet is not dead at this point--she is simply in a deep sleep brought on by a tincture given to her by the Friar, to fake her death to avoid her marriage to Paris.
The news of this plot was sent to Romeo, but the message boy did not reach Romeo in time. As far as Romeo knew, his love had passed away.
He wanted to be at her side in the end--for she had become his whole world, his only focus. He was so obsessed with his Juliet that he poisoned himself there on the spot, because he believed that a life in a world without Juliet was a life not worth living.
Dreadfully, sadly, as the poison works its' way through his system, our Sleeping Beauty begins to wake up, only to find that her Knight in Shining Armor has fatally poisoned himself just moments before her awakening.
This is an intense and tragic scene, with much build up and a great deal of emotion--grief, love, loss, confusion. So, just as Romeo could not live in a world without Juliet, Juliet cannot live in a world without Romeo. She takes his knife from his side and stabs it into her heart, falling upon the body of her love, dying for him as he died for her.