It's not ungrammatical, as other answers mention, but it is not what most people would say.
Using the Corpus of Contemporary American English, I get 2 results for "doing teaching".
The phrase that I would use is "I'm teaching"; "am teaching" gives 51 hits.
Both phrases use the present continuous tense. However, the first phrase is uncommon due to the horror aequi principle:
The horror aequi principle involves the widespread (and presumably universal) tendency to avoid the repetition of identical and adjacent grammatical elements and structures. — Gunter Rohdenburg, “Cognitive complexity and horror aequi as factors determining the use of interrogative clause linkers in English”, in Rohdenburg and B. Mondorf, Determinants of Grammatical Variation in English, 2003.
(There are some great answers about this principal here, including this one.)
In other words, it's not wrong because it's ungrammatical; it's "wrong" because it's in violation of the horror aequi principal.