Well, the answer to this question is going to depend quite a bit on what gender you are, unless you are willing to engage in some very authentic Shakespearian gender flexibility! Let us remember that boys played the parts of girls in Shakespeare's original plays. Having said this, I will give you one from Juliet and one from Romeo, and then you can pick which one you like.
My favourite dramatic monologue from Juliet comes in Act IV scene 3, which is the night before her wedding day to Paris. Juliet is left by herself to contemplate what she is going to do. She plans to drink the potion given to her by the Friar, but she doesn't know if it will work or if it might actually kill her. This is an impressive monologue because we see a very vulnerable Juliet who nevertheless decides she has to act alone, and thus gives us evidence of how she matures during the play.
For Romeo, I would pick his final soliloquy just before killing himself in Act V scene 3. This is clearly an intense moment, and is bittersweet in the way that it presents a reunion with his beloved wife, but it also represents a tragic twist of events, as Romeo kills himself just before Juliet wakes up. Hope these ideas help and good luck!