He asked how I was is the more usual idiomatic form. He asked how I am is a possibility, but it suggests a more continuous sense of "I am".

He asked how am I is not grammatical, punctuated like that. He asked "How am I?" is a soliloquy, in which he poses the question to himself.

He asked how I was is the more usual idiomatic form. He asked how I am is a possibility, but it suggests a more continuous sense of "I am".

He asked how am I is not grammatical, punctuated like that. He asked, "How am I?" is a soliloquy, in which he poses the question to himself.

He asked how I was is the correct idiomatic form.

He asked how I was is the more usual idiomatic form. He asked how I am is a possibility, but it suggests a more continuous sense of "I am".

He asked how am I is not grammatical, punctuated like that. He asked "How am I?" is a soliloquy, in which he poses the question to himself.