How to calculate probability of non independence using bayes theorem?
I think in practice you would make the assumption of independence in any case. Considering all the dependencies can get quite complicated very quickly and you'd need many observations. For example, in your example A=1, B=2 and C=2 was never observed,but you can't really tell, if that is because P(A=1 , B=2, C=2 | C1) is small and there is too little data, or if P(A=1 , B=2, C=2 | C1)=0. In many practical applications the approximation with independence introduces less error than not having enough observations for all possible events.

There is hybrid approaches, though, where you would model some dependence, e.g. tree-augmented naive bayes. These approaches usually perform better, but are computationally more complicated as well.