you will have a VERY hard time getting any kind of work.
1) as stated before you need a visa
2) why would someone hire you when you only plan on staying for 2 months?
3) by the time you found enough illegal tutoring, your 2 months would be finished.
4) there are enough professionals out there now, that this kind of backpacker mentality has gone the way of the do-do bird. it might have been possible back in the early 90s but no longer.
you might have more luck in taiwan or korea.