It is Romeo's birth that makes him a pawn of fate. His birth is the one thing he has no control over, and he was born into a family that was engaged in a longstanding feud with another family. It is essentially Lords Capulet and Montague's decision to continue the feud that leads to his death; however, Romeo's own decisions also contribute to his own demise.

For example, Romeo makes a choice against his better judgement to allow his friends to persuade him to crash the Capulet's ball. As he explains to his friends, he felt the decision was rash and actually had a dream prophesying that the events of the night would lead to his untimely death, as we see in his lines:

... for my mind misgives
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date
With this night's revels and expire the term
Of a despised life, clos'd in my breast,
By some vile forfeit of untimely death. (I.iv.113-17)

Had Romeo stood by his decision that crashing the ball was dangerous and rash, he never would have angered Tybalt and never would have caused the events leading to his own death, as well as Tybalt's, Mercutio's, and Juliet's.

Another choice Romeo makes that leads to his demise is to avenge himself on Tybalt for Mercutio's death. After Tybalt kills Mercutio, Tybalt flees the scene. Had Romeo fled at that moment as well, he would not have killed Tybalt, and he would not have been exiled, which are both events that lead up to Romeo's own death. However, instead, Romeo decides to stand his ground and seek revenge, which he gets to do the moment Tybalt returns. Romeo actually had no need to avenge Mercutio's death. Prince Escalus had already decreed that any more fighting would be punished by death; Tybalt was already doomed to die. Romeo could have made the more reasonable and rational decision to escape the moment and allow the law to avenge Mercutio's death. However, Romeo failed to make the more rational decision, which eventually leads to his own death as well.

Does Shakespeare present his characters Romeo and Juliet of the play Romeo and Juliet as responsible for what happens, or are they just the playthings of fate?

Shakespeare presents the fatalities of Romeo and Juliet as both a product of fate and of personal choices. On the one hand, Romeo and Juliet were fated to die due to the fact that they were born to parents of warring families. However, the fact that the families were warring was a choice made and carried out by the family members. Not only that, Shakespeare makes it very clear that both Romeo and Juliet had fatal character flaws of being impulsive.While the play opens by referring to the two "star-cross'd lovers" born of two enemy households, we soon learn that the Prince holds the households responsible for continuing the feud and is now threatening the families with death should they continue. Not only that, even Capulet declares that men his and Montagues' age should be able to keep the peace in the city, showing us that even he...