You have a fairly complicated history here. You can’t just tell them your medical history though. They need to know that if they allow you to return, you will succeed. Nothing you have said above shows that.
Your appeal needs to acknowledge that you:
-take responsibility for your actions. You can briefly state what the problem was.
-address HOW you will stop the same thing from happening again. That could include information about therapy or medication you will be regularly taking, along with lifestyle changes such as lower work hours, less partying, whatever it might be.
-specifics about how you plan to succeed going forward: “I will attend all classes, complete all homework, and study for all tests. I will go to professor office hours once a week. I will make a schedule for studying. I will join a study group. I will retake all failed classes. And so forth…”
-Reaffirm that you are committed to your major and the school and that you appreciate their time. Be sure to thank them.
Be concise, don’t add a lot of words that are just filler. Get to the point, which is convincing them that you can succeed. You’re off to a good start by coming here to ask for advice.
ETA: I recommend you get your psychiatrist to submit a letter also, detailing how you are addressing your mental health issues.