It seems if she 'took you back' like you said after mutiply times, to a girl, that means that she has feelings for you and wants to try again. (Alot of girls won't waste their time if they don't think the guy is worth it. By what you say she thinks u are worth it!) You mentioned you went out with her four times, have you known her longer then that? Do you feel comfortable with her knowning about the bipolar- that in my opinion is extremely important? How comfortable are you with her? The good thing is she 'took you back' so telling her may help her understand why your personality at times changes drastically. Is there a way you can share some details with her? It basically depends on how comfortable are you with her knowning. Also, how much do you want to let your guard down with her? You may be presently surprised and she's understanding with what you feel comfortable with telling her.
Good Luck!!! Let me know what happens!