How does Romeo find out that Juliet shares his feelings and what does Juliet seem most concerned about during the balcony scene?

Romeo has an idea that Juliet is interested in him in Act I, Scene 5 when they meet at the party. After all, she allows him to kiss her twice within minutes of first meeting. If that were not enough evidence for Romeo he overhears her speak his name as he hides under her balcony in Act II, Scene 2, the balcony scene. She says,

O Romeo, Romeo, wherefore art thou Romeo?
Deny thy father and refuse thy name,
Or, if thou wilt not, be but sworn my love,
And I’ll no longer be a Capulet.

Here, she not only refers to him as her love, but also wishes that they were not from feuding families. She urges Romeo to give up his name so that they can love each other without problems. 

At first, Juliet is concerned that Romeo might be discovered in the orchard and she warns him that he would be killed if her family found out. She says,

The orchard walls are high and hard to climb,
And the place death, considering who thou art,
If any of my kinsmen find thee here.

Romeo assures her that he would rather die than not be loved by her. Juliet is also concerned whether Romeo is really in love with her or is just leading her on. She asks him to swear his love. When he swears on the moon she takes exception because the moon is always changing (an interesting point here is that the audience knows of Romeo's mercurial nature as earlier that day he had been totally infatuated with another girl). She says,

O, swear not by the moon, th’ inconstant moon,
That monthly changes in her circled orb,
Lest that thy love prove likewise variable.

She is afraid that his love may be fleeting and that he will eventually change his mind. She also believes that maybe she has been too forward and wishes she had been more shy, but since her secret is out she pledges her love is not "light", but very serious. She says,

I should have been more strange, I must confess,
But that thou overheard’st ere I was ware
My true-love passion. Therefore pardon me,
And not impute this yielding to light love,
Which the dark night hath so discoverèd.

Finally, she thinks maybe they are rushing into things and that they should sleep on it and see how they feel the next day. She says,

I have no joy of this contract tonight.
It is too rash, too unadvised, too sudden,
Too like the lightning, which doth cease to be
Ere one can say “It lightens.” Sweet, good night.
This bud of love, by summer’s ripening breath,
May prove a beauteous flower when next we meet.

They agree to be married the next day, setting the stage for the tragedy to follow.