Hello, I'd like to help you with your question.
It sounds like your boyfriend might have a personality disorder. Blaming others, cheating in a relationship and being self centered all point to someone who may be narcissistic or even anti social.
It is good that you got out of this relationship and have moved on. He did
not seem willing to accept responsibility for his actions, especially cheating on you. And even though you gave him another chance, he still would not end the other relationship. That is a sign that he was putting his own needs first and not your needs or feelings. His request that you forgive him may be another way to try to manipulate you. So far, his behavior has shown that he sees everything from his own point of view and is not concerned with your feelings. Based on his past actions, you may assume that his asking for forgiveness is the same motivation.
It's not clear if he has Bipolar based on his behavior. Bipolar can cause erratic behavior. Here is a guide on Bipolar to help you:http://www.helpguide.org/mental/bipolar_disorder_symptoms_treatment.htm
Here is another site about personality disorders, which he may have based on what you have seen with him:http://www.nmha.org/go/information/get-info/personality-disorders
It is very positive that you have decided to move on with your life. There is nothing to gain for you by letting him know you have forgiven him. You can do that on a personal level for yourself so you can let go of the past, but letting him be involved in that process only feeds whatever need he seems to have by getting you to do this for him. It is probably better for you and your peace of mind to cut off all contact with him so you can remove yourself from the toxic environment he creates.
I hope this has helped you,