How is hate intertwined within Romeo and Juliet?

In Shakespeare's Romeo and Juliet, the theme of hate is most prominently seen in the feud that exists between the Capulets and Montagues. It is because of this ongoing feud—and no one remembers why it even started—that Romeo and Juliet not only meet and marry in secret, but also die. Their families will not let the past go, but insist upon feeding the hate that stands between them.
One of the play's themes is hatred.

Hatred breeds disaster, and the feuding families’ blind intolerance is punished with the deaths of their children.

This hatred is found at the beginning of the play. Although the Prince has outlawed any fighting associated with the feud, Capulet's servants are still prepared to do battle, most especially Sampson. While Gregory explains that it is only their quarrel because it is their master's quarrel, Sampson is ready to fight and face the hangman's noose.
Enter Benvolio and Tybalt. While Benvolio tries to stop the brewing fight from taking off, Tybalt (a real hothead) is ready to come to blows with Benvolio. He is concise about what (and who) he hates. He draws on Benvolio:

TYB:
What, drawn, and talk of peace? I hate the word As I hate hell, all Montagues, and thee. Have at thee, coward! (Act I, Scene 1, lines 65-67)

Old Capulet and Old Montague enter with their wives. Each man is ready to fight, and each wife tries to stop her husband from drawing his sword.
In Act III, Scene 1, Tybalt comes upon Mercutio while looking for Romeo. The two men throw insults at each other, and Mercutio is ready to cross swords with Tybalt. Romeo arrives and tries to make peace. Because Romeo previously attended the Capulets' ball uninvited, Tybalt is furious. Even though Capulet told Tybalt to leave Romeo alone, Tybalt announces that he is looking for and wants to fight Romeo. As they come face to face, Tybalt challenges Romeo:

TYB:
Romeo, the love I bear thee can afford No better term than this: thou art a villain (lines 59-60).

Tybalt does not "love" Romeo; he hates him. Romeo does his best to prevent violence. He asks Benvolio for his help and tells Tybalt that he (Romeo) has no quarrel with him. As Romeo tries to shield Mercutio and keep the sword fight at bay, Tybalt slides under Romeo's arm with his sword and kills Mercutio. Stunned with disbelief and self-hatred for not standing up to Tybalt (even though he did it for Juliet's sake—Tybalt is her cousin), Romeo pursues and kills Tybalt.
The hate established in this scene initially involves Tybalt and Mercutio. When Romeo arrives, it shifts, but Tybalt takes a cheap shot and kills Mercutio. Romeo's anger gets the best of him, and he exacts revenge for his friend's death.
As the end of the play approaches, Paris's hate for the Montagues is obvious. Paris is infuriated that Romeo killed Juliet's cousin, Tybalt. When Romeo arrives at Juliet's burial site to see the woman he believes is dead, Paris (who is already there) attacks. Romeo tries to warn Paris off, but the other man cannot be reasoned with, and Romeo kills Paris to defend himself. Romeo, still believing Juliet is dead, poisons himself and dies. When Juliet wakes and sees Romeo dead, she stabs herself with his dagger and dies. 
An alarm goes out in grief over Paris, Romeo, and Juliet. The Capulets and Montagues meet the Prince. Lady Montague has just died out of grief over Romeo's banishment. Now the parents of the young couple learn what has happened—all because of their feud. Capulet offers his hand to Montague, and the two men make peace. Montague says he will build a statue to honor Juliet. Capulet will do the same for Romeo, noting that the statues are "Poor sacrifices of our enmity" (Act V, Scene 3, line 316).
In other words, the statues mean little in light of what has been lost—Romeo and Juliet, their beloved children—because of the hate that existed between these two men and their families.