Explaining to the people in your life that you have Bipolar (I assume you have been formally diagnosed) may provide some explanation for them of your behavior and problems, but I don't think your husband will accept this as an excuse for your cheating.

Also you are very quick to advocate for your lack of responsibility due to your bipolar, most mentally ill people are not so skilled at persistently asserting that they are not responsible for their actions due to mental illness!

The truth is there are some lines that if you cross (whether because of bipolar or not), that people will not accept you afterwards. I don't know if your husband can possibly forgive you and move on with your relationship, but your continuing to see this other man- it is asking for your marriage to be over for good!