Given the fact that Facebook is a public forum proceed with caution when you are interacting with the person you have romantic interest in.
People find it much easier to be more flirtatious, suggestive, daring and provocative online than they do in real life. Be aware of that; and don't do or say anything online you would have trouble doing or saying offline.
Keep the first few exchanges flirtation-free.
Meaning, take it easy on compliments and don't get overly personal. It's one thing to compliment someone's photo, it's quite another to say "you'd look so much better in that picture with me on your arm. "
Before getting a tad bit more personal, scope out the person's profile and determine whether or not they are involved with someone else who can view your comments.
If you determine that they are unavailable, back off. Don't post suggestive comments on their wall, do not discuss their significant other, simply let it go and move on. If you don't, you run the risk of a confrontation with the person you have interest in, their significant other or both. Think before you speak; type that is. You don't want to offend anyone or make yourself appear desperate.
Don't stalk your love interest on Facebook.
Post an occasional interesting or complementary comment on their wall, but refrain from commenting excessively. You won't come across as relationship material if it appears as though your entire life is on Facebook. After all, most people are not interested in cyber relationships, they are interested in real life ones.
Don't send requests to people he/she knows and you don't.
Don't try to befriend them. You wouldn't do this in real life. At least I hope you wouldn't, and I assume you would wait until you get introduced. . . Even then proceed with caution.
As soon as you've established some rapport and mutual interest is clear, take the relationship off Facebook.
Exchange email addresses, phone numbers and communicate the old fashioned way. . . the best way if you ask me. Your relationship is in its infancy and it will take effort to build; subjecting it to public scrutiny can seriously hinder its development and even cast you in a negative light. Good luck.