Question 1: I'm doing very well. How about yourself?
Question 2: I'm doing very well. How about you?
Please explain the difference if both questions are correct.
'How about you?' - can refer to one or two people so that would be the reason why I'd want to use 'yourself' or 'yourselves'.
I personally use the colloquial usage if I want to differentiate between one and many people and it is accepted where I live as in:
'How about yourself?' or 'How about yourselves?'
You could say
'How about you three', 'how about you in front of me', 'how about you lot'.
but that all sounds a bit rude because it doesn't address those people as people! (you could be asking your robots)
Possibly better grammatically might be
'How about you, yourself?' or 'How about you, yourselves?'
but that is just a bit longer and the 'you' just seems a bit pointless - the brain can work out without any extra effort what you mean. So, I'd use 'you' if there is no ambiguity between the person I'm addressing and others and 'yourself' or 'yourselves' without the 'you' if there is ambiguity.
The sentence with "yourself" is not, strictly speaking, wrong. It sounds a bit folksy and colloquial, like the person who is saying this is old friends with the person he is addressing.
The thing is, these formulas that are used in greetings are set phrases. Even if an alternative is not completely wrong, it's best not to insist on using them.
A better use of "yourself" might be in an exchange like this,
– How are you?
– Fine. And yourself?
This is a shortened version of the proper phrase. Again, colloquial.
If you are asking about subtexts of meaning, using yourself underscores that you are reversing the question from you back to the person who has asked first.
In the context of both of the questions, both seem to have the same meaning. But there's a subtle difference in you and yourself depending on the situation. 'Yourself' is used reflexively to reflect back to 'you' or 'me' as the subject. E.g., You picked it up yourself. Hope this helps.
How about yourself is grammatically incorrect. When using reflexive pronouns, you should use a subject at least once, although you can have an implied use to e.g.. You did that to yourself or (You) Look at yourselves. It is grammatically correct to say How about you, yourself, but that's pretty clunky.