Most directly, the "hateful relationship" of the two families causes the death of Romeo and Juliet. The prologue tells us this:
From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whose misadventur'd piteous overthrows
Doth, with their death, bury their parents’ strife.
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,
Is now the two hours’ traffic of our stage;
It is the death of the children that ends the "parents' strife." Nothing else could make the rage go away.
How does it all come about? When Romeo and Juliet meet, there is an instant connection that is shortly built to love in their minds. They are both distraught to discover that the other one belongs to the "enemy." However, Romeo sneaks into the garden of the Capulets and meets with Juliet at her balcony. This fuels their attraction and love, and makes them eager to overcome the obstacle of their families. However, there is no way for them to meet. Juliet, as a young woman in this time period, does not have the freedom to move about. Therefore, they jump to marriage. Once married, their families can't intervene.
Or can they? Tybalt, upset that Romeo showed up at the Capulets' ball, confronts Romeo. This is the hatred between the families again getting in Romeo and Juliet's way. In the confrontation, Tybalt kills Mercutio and Romeo kills Tybalt in revenge. Romeo makes a clear choice to allow the families' conflict win out over his new bride:
Away to heaven respective lenity,
And fire-ey'd fury be my conduct now!
Romeo is sending away the "lenity", or gentleness, that had him ignore Tybalt's insults, knowing that Tybalt is a cousin to his new wife. But fury now wins out, and he kills Tybalt, causing him to be banished. This sets into action the chain of events that leads to both characters committing suicide.