How could Fate be seen as playing a part in Romeo and Juliet's death?
I think that the Prologue is probably the best starting point in examining the role of fate in the two lovers' deaths. Consider that Shakespeare himself opens with fate's hand being present in the drama:
From forth the fatal loins of these two foes(5)
A pair of star-cross'd lovers take their life;
This brings out the idea that the deaths were fated because Romeo and Juliet entered into a configuration that was outside of their own control. The "ancient grudge" to which there is reference in the opening lines of the Prologue indicate that this set up was larger than the two lovers. The very idea of fate is present in the fact that both young people really had no chance of controlling anything in their world other than their feelings. They could not alter this set up, nor could they effectively abolish it. They were pawns, for lack of a better word, within it. If the argument about their deaths being fated is going to be advocated, I think you would have to look at the plans that both lovers make to being together as being trapped in futility. There is an undercurrent that while both of them want to be together, something, some type of force, is going to inject itself into the discussion to prevent this from happening. Perhaps, this feeling is developed from the Prologue's warning, but it settles into the characters themselves. Before Juliet take the potion, she goes through all the various calculations and possibilities that exist, almost trying to rationalize the whole thing, but understanding that she is unable to do so. The fact that she would contemplate all of these "mischances" helps to reveal that at her most critical moment, she feels the weight of fate upon her and tries to shake it off, recognizing that it is too powerful. If this argument is going to be made, I would look at these areas as starting points for its substantiation.