Two Questions in One Sentence
Is it correct to ask:

Do you care about how am I?

or should it be:

Do you care about how I am?

which one is correct? my friend is insisting that the first one is correct.

Question

Two Questions in One Sentence
Is it correct to ask:

Do you care about how am I?

or should it be:

Do you care about how I am?

which one is correct? my friend is insisting that the first one is correct.

Answer

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 ?".

Answer

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

Answer

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

Answer

I think the second one is correct

ご自宅で快適に語学を学べるチャンスをお見逃しなく。経験豊富な語学講師陣の中からお選びいただき、今すぐ最初のレッスンにお申し込みください！