First, the lovers face the terrible difficulty of being from families which have been feuding for quite a long time. The Capulets and Montagues bear an "ancient grudge" against each other, and this will make it very difficult for their children when they fall in love (Prologue, line 3). Further, the speaker of the Prologue says, "From forth the fatal loins of these two foes / A pair of star-crossed lovers take their life" (5-6). Thus, not only will Romeo and Juliet have to contend with their families' rivalries, they also have fate or destiny to contend with as well. They are destined to take their own lives.
Further, when Tybalt slays Romeo's best friend, Mercutio, Romeo kills Tybalt. Tybalt is Juliet's cousin. Therefore, besides whatever it was the two families were feuding about before, there is this fresh upset to anger them anew. This increased and renewed animosity poses a great deal more trouble for the couple. In addition, as Mercutio lay dying, he curses them both, saying "A plague o' both your houses!" (3.1.111). So, on top of their families and fate being against them, the couple has now been cursed to suffer by Romeo's best friend.