It sounds like you’ve already had more to handle than someone who is only 13 should. You didn’t mention the kind of health problems you’ve had. Chronic or serious illness is enough to get anyone down. Sometimes too there are long term effects of the medicines or anesthesia (if you had to have surgery) that look and feel like depression. If that weren’t enough, for reasons neither you nor I understand, the very people who should be a source of warmth and comfort are instead blaming you and labeling you in negative ways. It must hurt terribly.
I don’t know what’s going on with the family. I do know you can’t manage this on your own. You need someone who can help your family hear you. Ideally, you would all get into family therapy. But if the family is the kind that would never consider it, do at least enlist your school guidance counselor or a trusted teacher or some other adult who is willing to help sort things out. You could also ask your doctor for a few minutes in private to explain what is going on and to ask for some help.
Meanwhile, develop an active life with supportive friends outside the family. If you feel good about yourself, the accusations and criticisms won’t hurt quite so much. You will know that you’re okay even if you don’t get the support and validation you should be getting from the family. Join a team or club. Eat right, get at least 8 hours of sleep a night, and make sure you get some exercise every day to combat the depression. Turn yourself into the kind of guy that other kids respect. Good friends are the best medicine for a disappointing family.
I wish you well.