Who is most responsible for Romeo and Juliet's deaths?
There is a great deal of debate over this very subject. Lets start with a few possibilities, and then examine what Shakespeare himself said.
First, one must consider Romeo and Juliet themselves. They are impetuous teenagers. No one was forcing them to rush into marriage, and no one forced them to take their own lives. They behaved immaturely and irrationally, and did actually commit suicide.
Second, there are many arguments for the Friar and the Nurse. As adults, and as co-conspirators, they had many opportunities to come forward and "turn the kids in". If not that, they could have at least counseled Romeo and Juliet against their choice of marriage, recognizing that it was too problematic and certainly too soon. The Nurse is the only one to encourage Juliet to leave Romeo's path, when she encourages Juliet to marry Paris. But bottom line, these adults weren't guiding their charges very well.
An argument could also be made for Mercutio and Tybalt. Had either one been able to control their temper, as Romeo asked both to do, then the events would not have been set in motion that led to Romeo and Juliet's death.
However, Shakespeare goes to great length to tell the audience that it is the parents, the families at large, and their feud that is the real cause for the teenagers' deaths. See these lines from the prologue:
The fearful passage of their death-mark'd love,
And the continuance of their parents' rage,
Which but their children's end naught could remove,
Their love is death-mark'd because of their parents' rage, as proven by the fact that the rage only ended when Romeo and Juliet's lives ended. Then, in the last scene, Shakespeare has the prince lay the blame on Capulet and Montague (and, incidentally, himself):
Where be these enemies?--Capulet,--Montague,--
See what a scourge is laid upon your hate,
That heaven finds means to kill your joys with love!
And I, for winking at your discords too,
Have lost a brace of kinsmen:--all are punish'd.
The Prince says they are all punished for the feud by the deaths of those they love. They are responsible.
So, there's a few options for you. See which one makes the most sense for your interpretation.