The first one is definitely incorrect. The second one is correct but sounds unnatural. It would sound more natural/native English to say "Do you care about me ?" or, "Do you care how I am ?". You could also say, "Do you care how I feel ?".

Also: "Do you care [about] how I'm doing?"

It is the second. As Bookworm says, it might be more idiomatic to omit "about", although for me it sounds quite natural either way.

I think the second one is correct