The prologue is the very first sign from Shakespeare that indicates it is fate that Romeo and Juliet kill themselves. This is clear in line 6, when it says, "A pair of star-crossed lovers take their life." Calling them "star-crossed" is his way of showing that it is not their actions that lead to their death. It was already planned that their paths would cross before they met, written in the stars, possibly as a way to end the feud between their families.
The next quote that demonstrates this is in Act I, Scene iv, lines 107-114, when Romeo and the Montague boys are about to crash the Capulet party. Romeo says:
"I fear too early, 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 despisèd life closed in my breast
By some vile forfeit of untimely death.
But he that hath the steerage of my course,
Direct my sail."
He is basically saying here that he has a feeling fate has brought him to this party and that his own death will be caused by the events of this night. Here he also gives up his own free will, saying that whoever is in charge of his destiny should "direct his sail," or lead him to whatever his fate may be.
In Act IV, Scene iii, there is another quote that proves Romeo and Juliet have no free will and are already destined to die. In lines 53-56, Romeo is exiting Juliet's window after he has killed Tybalt and they have spent the night together. Juliet looks down at him and imagines that he is in a tomb, saying:
"O God, I have an ill-divining soul.
Methinks I see thee now, thou art so low
As one dead in the bottom of a tomb.
Either my eyesight fails, or thou look’st pale."
"And trust me, love, in my eye so do you."
Some part of both characters already knows that they are fated to die, and Shakespeare has them say lines like this to make the situation all the more tragic, showing that the characters have no free will and their fate is already written.