It is hard to say that any individual character in Romeo and Juliet displays inherent evil, at least not to the extent of other evil Shakespearean characters such as Iago or Richard III. No one plots to destroy anyone's lives. Rather, unfortunate events and misunderstandings lead to the demise of the two young lovers. Even the violence that dominates Act I, Scene 1 is not borne of true evil. It is simply reckless mischief on the part of the Montague and Capulet subordinates.
Some may point to Tybalt as evil, but it is hard to read that much into his character. The fact that he is quite well loved by the family and especially the Nurse tells us that he has redeeming qualities. Some may also say that Capulet displays elements of evil when he is berating Juliet over marrying Paris, but, in reality, we simply find a father who is trying to do the right thing for his daughter.
There are, however, two expressions of evil that can be pointed to in the play. In Mercutio's Queen Mab monologue in Act I, Scene 4, he suggests that Queen Mab puts evil thoughts into people's minds such as a priest who is greedy or a soldier who dreams of killing his enemy:
Tickling a parson’s nose as he lies asleep,
Then he dreams of another benefice.
Sometime she driveth o'er a soldier’s neck,
And then dreams he of cutting foreign throats,
Of breaches, ambuscadoes, Spanish blades
When we first meet Friar Lawrence
he is in his garden picking flowers and weeds to make potions and medicines. The scene serves mainly as foreshadowing
for later in the play when he gives Juliet a vial of sleeping potion to fake her death. Among these weeds the Friar finds both healing medicines and poisons. He compares this to man's nature. Within the earth there are things that can be of benefit and things that carry great evil, not unlike man's nature. The Friar says,
Within the infant rind of this weak flower
Poison hath residence and medicine power:
For this, being smelt, with that part cheers each
Being tasted, stays all senses with the heart.
Two such opposèd kings encamp them still
In man as well as herbs—grace and rude will;
And where the worser is predominant,
Full soon the canker death eats up that plant.
Shakespeare is making a comment not only on mankind, but also on the feud between the Montagues and Capulets. Even though many good people are involved, the feud itself is essentially evil and leads to several deaths.