How do you reply to how are you doing well or doing good?

That depends on how you're doing!

I think you want to know if the answer should be, "Good," or "Well." Since you need an adverb to tell about the verb doing, your answer should be "I'm well. How are you?" When you say that you are good, it means you are a good person. It doesn't tell how you are doing since good is an adjective which modifies a noun, not a verb. The Word well is an adverb which modifies a verb, in this case the verb doing. "I am doing well" is correct. "I am doing good" means I am doing good works or I am doing good things or I am doing good in the world . . . (You get the idea.) It doesn't describe HOW you are doing. For that, you need to use the Word well.

Of course, in everyday conversation, we do not always follow the rules of standard English, and our friends forgive us!