Romeo, throughout Romeo and Juliet, is characterized as having a brash and tempestuous personality, and this emotional turbulence certainly contributed his and Juliet's deaths.
Possibly the most critical turning point in the play's action is when Romeo kills Tybalt, spurred on by Mercutio's death. This turn of events causes Romeo and Juliet's situation, which was already precarious, to deteriorate drastically. With Romeo killing such a prominent Capulet, the feud between the two families is more heated than ever, and Romeo's banishment means he and Juliet will be separated. It is in this context that Friar Laurence suggests the plot to fake Juliet's death.
This is not the end to Romeo's rash, emotionally-behavior. After learning of Juliet's supposed death, Romeo's immediate response is to buy poison and commit suicide. Had he consulted Friar Laurence or hesitated even a little bit more before acting, the play's resolution would have been drastically different. Juliet awakens from her induced slumber only moments after Romeo drinks the fatal poison, meaning that if he had waited just a little bit longer, they may both have lived.
At the same time, however, it's important to recognize that there were other factors contributing to this tragedy that were well outside Romeo's ability to control. Fate and chance, for example, play a critical role in Romeo and Juliet's tragic demise. If Friar John had been able to successfully deliver the letter explaining the plan to Romeo, if Friar Laurence had reached the tomb in time, if Romeo had simply been delayed a little bit on his return to Verona—if any of these conditions had changed, the resolution would have been different.
Romeo's rash actions from the beginning to the end of the play lead to disastrous consequences, including his own death and Juliet's suicide. When Romeo first falls in love with Juliet, he is on the rebound from being rejected by Rosaline. (It's always wise to have a cooling-off period between relationships to make sure that one's emotions aren't running wild.) Romeo sees Juliet and immediately proclaims her to be the loveliest person he's ever seen. Does this warrant rushing into a relationship that will certainly be problematic? Romeo doesn't stop to think of that—he lets his eyes rule his brain.
Within minutes of meeting Juliet, he has kissed her twice. Jumping into a physical relationship, especially with a young and inexperienced partner, before getting to know the person is also unwise. Throwing caution and clear-mindedness to the wind, Romeo agrees to marry Juliet on the very night they meet, and he has soon talked Friar Lawrence into performing the ceremony.
All that hastiness might have been overcome, but then Romeo makes everything exponentially worse by fighting and killing Tybalt. His motivation for murdering Juliet's cousin is not at all noble—he says that love for Juliet has made him "effeminate." In other words, if he doesn't avenge Mercutio, he'll be known as a sissy. Romeo is lucky indeed to be only banished, not executed, for his crime against Tybalt.
Later, Romeo returns to Verona to visit Juliet, whom he believes to be dead, and to kill himself with poison he purchased from an apothecary. At the tomb, he kills Paris and then himself. When Juliet, who was not dead but under the influence of Friar Lawrence's potion, wakes to see her husband dead, she stabs herself. Romeo's rash and foolish actions result in his own death and encourage Juliet to follow his example of suicide.
There are a number of reasons why one could believe that Romeo is responsible for his death and Juliet's death. First of all, Romeo should never have attended the Capulet party in Act I, Scene V. As a Montague and sworn enemy of the Capulets, he was not invited to the party. Had he not ignored the rules and gone to this party, he would not have met Juliet and fallen in love with her, thereby saving both of their lives.
Secondly, Romeo is also responsible because he did not heed the advice of the Friar. Remember that in Act II, Scene III, the Friar tells Romeo not to rush into a marriage with Juliet because he has only just got over Rosaline. Instead of listening to the Friar, Romeo continues to press the marriage. Had Romeo and Juliet not married, Juliet would not have been forced to drink the potion to prevent her marriage to Paris. As we know, it is this sleeping potion which made Romeo think that she was dead and take his own life.
Finally, the killing of Tybalt also played a role in his responsibility. Had Romeo not killed Tybalt and been banished, he could have stayed in Verona and possibly healed the rift between the families. Instead, the murder of Tybalt made the Capulet family hate him even more.
I believe the main reason that Romeo is responsible for his death as well as Juliet's is purely and simply his rashness and impatience in nearly every event of the drama. He rushes into his relationship with Juliet, which, by the way, I believe, he sort of urges her into, then the marriage. He then kills Tybalt in the heat of the moment and gets himself banished. Now the ultimate way in which he causes both of their deaths is in his failure to wait or take time to think before taking his own life when he finds Juliet in the tomb. His immediate reaction is to drink the poison; if he had waited a mere matter of minutes, she would have awakened and they could have gone away together. However, Romeo, always the impetuous hothead that he is, refuses to wait for anything; he can't take anything that remotely resembles pain and chooses death instead. This of course leads Juliet to take the same course of action. Hope this helps.
Well, one idea to consider is that if Romeo had not killed Tybalt, he would not have been sent into exile, and thus neither of them would have died. You could also consider the idea that ROmeo acted rashly by buying poison to kill himself after hearing of Juliet's death. If he had waited a few moments, he would have realized that she was just asleep.