One comparison that can be made between Brave New World and Macbeth is in the treatment of fate and destiny and the characters' reactions to their fates.

Bernard Marx and Helmholtz Watson of Brave New World and Macbeth feel their fates are determined by elements outside their control. Whereas it is scientific deceit which causes the lives of Marx and Watson to become tragic, it is the accepting of fate in the mind of Macbeth that leads him to be influenced by the preternatural world which later becomes his nemesis.

In Brave New World, in one of his first dates with Lenina, Bernard shocks her when he tells her that he enjoys the sunset and likes to be alone and feel more alive and individual:

"It makes me feel as though …" he hesitated, searching for words with which to express himself, "as though I were more me, if you see what I mean. More on my own, not so completely a part of something else. Not just a cell in the social body."

Like Bernard, who feels spiritually confined in his society, Helmholtz Watson, who is an Alpha Plus and of the highest intelligence in the New World, wishes that he could do more than write snappy slogans designed to promote the values of his society. But when John the Savage reads Shakespeare to Helmholtz, he realizes that he has been limited in what he can intellectually understand because of his society's conditioning of him.

Although he has not been pre-conditioned in his world, Macbeth allows himself to become susceptible to the designs of the preternatural world with the appearance of the three witches, who make predictions of his future. Influenced by this supernatural world, Macbeth allows himself to be led down a fateful path when he confuses reality and fantasy as he is driven by his ambition. 

While he is less a product of a controlled fate than Bernard Marx and Helmholtz Watson, Macbeth does become controlled by the fate predicted by the witches because in his "vaulting ambition" he accepts reality and fantasy as equal so that he can be king of Scotland. Indeed, it is his fatal confidence in the witches' predictions that later return Macbeth to the battlefield in the end, only to be finally defeated by his fate:

Accursed be that tongue that tells me so,
For it hath cowed my better part of man! (5.8.17-18)

Destiny, whether scientifically planned or one influenced by the supernatural world, affects the lives of the characters of both Brave New World and Macbeth.

The biggest comparison between the two would be the theme of knowledge vs. ignorance.  In Brave New World, the citizens are denied knowledge so that they will not be tempted to rebel.  The belief is that in ignorance, they are unaware of what they are missing.  This applies to Macbeth because he had too much knowledge.  The witches gave him so much information, he began to act in ways unusual for him and unproductive in order to speed up the events that were supposed to happen to him.  In he had lived in a Brave New World, he would not have had the knowledge and so would not have become a murderer.

This question has already been answered on eNotes.  Here is a link for you:  http://www.enotes.com/macbeth/q-and-a/whats-some-comparison-macbeth-brave-new-world-14363

Macbeth and Brave New World have the following in common:

• A tragic hero: Macbeth is ruled too much by ambition and the occult; John is ruled by his inability to reconcile the past and present worlds.  Both die as a result of these tragic flaws.
• Suicide as a mark of guilt and fateful choices: Lady Macbeth kills herself because of the guilt in Duncan's murder; John kills himself because of the guilt of overindulging in soma and orgy-portgy.
• A conflict between the natural and supernatural / unnatural world: Macbeth subverts the natural order by using the witches (supernatural) as advisors; John escapes the natural world (the Savage Reservation) and is thrown into the unnatural world of the Brave New World.
• A caste/class system in which women, in particular, are victimized: In Macbeth, Lady Macbeth must live vicariously through her husband.  The witches are the bottom of the social ladder.  Lady Macduff, too, is powerless.  In Brave New World, the women, Linda and Lenina in particular, are playthings to men.  They are second class citizens (Betas to the male Alphas).
• A subversion of family values and education of children: the Macbeths do not have children and would make terrible parents.  In the play there are bloody children (Macduff), the murdering and attempted murdering of children (Banquo's and Macduff's sons), and Lady Macbeth's admission that she would dash her baby's brains out.  In Brave New World, children are unwanted naturally; instead, their test-tube equivalents are brainwashed to hate education and nature.  Both authors depict dystopias in which children are unloved and unwanted.