Since great is an adjective, "I'm doing great" seems to be incorrect. It should be: "I'm doing (adverb)." You could say "I'm doing well."
Could you also say “I’m doing greatly”?
Most adjectives that describes the perceived ‘quality’ of an action carried out by someone, or their state of being, correspond to identical adverbs. Examples include such adjectives as ‘fine’, ‘good’, ‘bad’, ‘great’, ‘fantastic’, ‘horrible’, ‘okay’, etc.
Some, but not all, of these adjectives also have adverbial counterparts, whether formed by regular -ly derivation or otherwise, that are not identical to their adjectival forms. These are mandatorily used when describing degree/manner, rather than judging quality:
(Some dialects would be okay with using an adjective in the last example, but as far as I know, “Wages have great increased” is ungrammatical in every dialect of English, so the order of the constituents also plays a role here)
In the case of ‘good/well’, there is the additional problem that ‘well’ is both the adverb that corresponds to the adjective ‘good’, but also an adverb meaning “in good health”. This is part of the reason why so many know-it-alls insist that sentences like “I’m well” or “I’m doing good” are incorrect: they fail to realise that ‘good’ is an adverb and ‘well’ an adjective as well as vice versa. Moreover, the quality vs. degree/manner dichotomy does not fully apply to this particular pair either, further compounding the issue:
– are both equally valid, both when describing one’s current state of health (quality) and when answering a question on how one is getting along with a task (degree/manner).
Some verbs (doing, going, feeling) act as copular, or linking, verbs when used this way...you are basically saying "I am great". Saying "I am greatly" is obviously wrong. See http://www.usingenglish.com/glossary/copula-verb.html
"Doing" to me is an edge case here, but I'd argue for it being capable of being used as a copula, based on vernacular usage. If someone asks,
You most commonly hear (SE USA) "good", which implies "I'm doing good". (copula)
The more formal response is "well"/"I'm doing well". (non-copula)
You could, but people would look at you funny.