Both sentences are incorrect.
I asked him how are you?
has to be either:
I asked him how he was. (Indirect question)
I asked him, "How are you?" (Direct question)
I questioned him how are you?
has to be
I questioned him about how he was.
But this is a strange usage. When you question: VERB, def. #2 someone, it's usually about a crime, an accident, or some other significant event, and in that case, the right word is usually interrogate. The person being questioned is either a suspect, or a known perpetrator, or a witness who can provide material evidence, character evidence, or background information about the event.
The difference isn't that one word is more or less polite than the other, but that they are used to say two different things.