*I apologize in advance for typing too much.*
I suggest that you read the play and grasp the plot and the characters before you try to base a story on it. I LOVE this play. Every time you read it, you read something new. Did you know that the first time Romeo and Juliet meet is a sonnet, with the characters physically moving closer together with each stanza and then kissing on the final couplet?
While you've got to love the bard's fabulous writing, I love his cast of characters even more. Everyone knows Romeo and Juliet, but who knows of Mercutio and Romeo, or Rosaline and Romeo, or Tybalt and Lady Capulet? If I were you, I'd write a story from the POV of one of the supporting characters. Some interesting characters that you might want to include in your story:
The Bawdy Nurse: She is in constant competition with the mother - for Juliet's affection, loyalty, and obedience. She is basically a mother without the status of one, which adds pathos to what would otherwise be a merely amusing character. This motivates some very funny lines as well as some questionable actions and advice on the part of the bawdy nurse - and boy is she bawdy! think about how she might feel after her failure to provide real guidance ends in her charge's death. Oh - and, consequently, she'd also be out of a job.
Mercutio: Romeo's best friend, Mercutio is a jealous, outgoing character. He is very popular. Everyone loves Mercutio, and yet he is still deeply troubled by something that makes no sense to the other characters. In the original play and especially in Baz Lutherman's modern adaptation, you also find lines that might lead you to suspect that his interest in Romeo is more than friendly. Mercutio is dead by the end, but you may want to explore the original plot from his point-of-view, with Romeo as the object of desire and Juliet as a threat and vixen.
Benvolio: Benvolio is the voice of reason in an unreasonable world. Whenever a character proposes peace, suggests that a character take a moment to think before he/she acts on emotion, or warns of consequences, it is most often Benvolio (although the prince and the priest perform these functions when Benvolio is off stage). Unfortunately, he is most often ignored and also often drawn unwillingly into intense emotional situations in which he always attempts to calm the situation or put things to right. In the aftermath, Benvolio would probably inherit Romeo's birthright. It would fall to him to restore peace and order to the house of Montague.
Tybalt: Tybalt is all spleen, and therefore stands in direct contrast to Benvolio, as both are characterized as the most influential among their respective masters' men. Tybalt seems entirely devoid of reason. Whenever a fight occurs or harsh words are exchanged, Tybalt is most often the instigator. As a fun side note, you might find lines in the play that suggest that he was having an affair with Juliet's mother. :-) Again, he is dead by the end, but you might want to explore the original plot from his point-of-view.
You might also enjoy playing around with Juliet's licentious father or with the ethical dilemma faced by the priest when he decides to assist Juliet in deceit of the first order in order to prevent her threatened suicide - and then fails. Another character who is not often touched upon (because she does not appear in the action of the play) is Rosaline, Romeo's former love who broke his heart when she decided (or her parents decided, that is not clear) to become a nun. Presumably, she is at the Capulet dinner, and so she has the chance to witness her love gravitating toward another and then dying for this new love. Rosaline could take it as proof that her decision to live chastely is correct, or she could feel as though the entire situation is her own fault, and then spin out of control. The possibilities are endless. Enjoy!