Flat characters are those who remain unchanged through the course of a play and are 'one-dimensional' - they have only one perspective, quality or point of view.
I would describe Paris as a flat character. As suitor to Juliet he is a polite and courteous gentleman. As befits his social position as kinsman to Prince Escalus, Paris is an equitable match for Juliet. There is no reason to suggest that there is any other path for Juliet other than to accept his proposal. He does not change in his demeanour and is loyal to her even after death. In the final scene he takes flowers to her grave and vows to return each night to her -
Sweet flower, with flowers thy bridal bed I strew
(O woe! thy canopy is dust and stones)
Which with sweet water nightly I will dew;
Or, wanting that, with tears distill'd by moans.
The obsequies that I for thee will keep
Nightly shall be to strew thy grave and weep.
"Flat" characters are characters which don't change in the course of a plot. They are usually minor characters, used for plot purposes such as moving the action along or revealing secrets. In Romeo and Juliet, there are some flat characters that come to mind.
The nurse can be considered a flat character. Her purpose in the play is to help Romeo and Juliet get married. She is Juliet's closest confidant, and she meets with Romeo on her behalf to arrange the marriage time and place. But she is more comic relief than a fleshed-out character.
Another flat character is Benvolio. He is the same throughout the entire play, solid, temperate, and loyal. The irony is that others accuse him of being hot-tempered and wild, when they are the hot-tempered ones. This is a classic Shakespearean character type.
A way in which you can show these flat characters through text is by comparing early and later dialogue. For example, compare Benvolio's reaction to the first street fight and his attempt to stop the deadly duel between Mercutio and Tybalt.