It's not "grammatically correct" because it's missing the verb. But yes, native speakers love to squish up and stretch the language (I think that may apply to any language?). It's fun! There's no "perfect" pronunciation of this - the speaker's accent always takes over. Couple of tips with using this: first, use it only with friends (like Joey! ha), and ONLY if they've used the same phrase already, or in the past. Second, realise that often this phrase (plus a lot of slang) is said jokingly - the assumption being that you do know the proper grammar, of course. ;)

How are you doing? Native American pronounce it in a sound linking form How_er you doing, the "w" is linking to er, thus it sounds like how you doing.

Informal - formal

How do you do? or How are you doing? are correct. The first is the most used

"How you doin'" is a saying that you hear on the Wendy Williams Show, that a lot of her fans (mostly black women) will say. It is gramatically incorrect, but it is also a common phrase.