Both are okay. "Let me explain about my family" is better. But it would be more natural to say "let me tell you about my family." Or if there is some unusual characteristic about your family, you could say, "let me explain something about my family." What is the context?

You are right. Thank you Kareem, my example wasn't good to ask my question.

it doesn't seem natural. i think it's better to say "let me tell you about my family "