Effect of halogen atom on Markovnikov's rule I know that the Markovnikov's rule is because of the stability of carbocations. If a halogen atom (electron-withdrawing group) is directly attached to the carbon atom carrying a positive charge, would it destabilize the carbocation (for example dichloroethene reacts with HBr)and 'break' the Markovnikov's rule?
Markovnikov's rule is just a simple rule for predicting the structures of  alkenes in some reactions.
It is not advisable to apply this rule to all cases,it is often helpful to check the relative stability of carbocation intermediates.

In your question the carbocation will be stabilised by +M 
effect of halogen as halogens have lone pair of electrons.