Summarize:

If you don’t know the girl very well, it makes sense to first interact with her publicly on her wall. This type of interaction is a bit more casual, but whatever you say or share may be subject to the comments and likes of others, so keep that in mind. To get a conversation going, try making a good-natured joke about something she’s posted. This shows that you are engaged with what she’s interested in and that you have a sense of humor.  Keep your jokes clean and don’t try to tear down any other comments with yours. If she likes or comments positively on your post, that is a good sign. You could try progressing to a private message, if she’s receptive to your wall posts. Photos are a good way to reinforce a connection with someone you like. If you have photos together from a day at work or school trip, you could post them and tag her. You could also share a funny photo that is not of the two of you but appropriate to a post she makes, such as a meme, to make her laugh. If you and the girl you’re trying to chat with have any inside jokes, post one on her wall. Inside jokes are good because they don’t tire out quite as easily as other jokes. Posting one (even if it was created in a larger group) will make her smile and cement a bond between you. In all your Facebook wall interactions, strive to be kind. It can be hard to read sarcasm online. If your tone is difficult to discern, she may interpret your posts as critical, even if that’s not how you mean them.  Phrases to avoid might include, “I’m not rude; I just say what everyone else is thinking.”
Make a joke on her wall. Share pictures on her wall. Reinforce inside jokes when possible. Be kind.