In fact, every non-empty perfect subset of the real numbers has the cardinality ℵ of the continuum. Thus, a perfect set cannot be a counterexample to the continuum hypothesis. This probably assumes the axiom of choice; but this is usually assumed whenever discussing CH and GCH.
Georg Cantor's interest in the continuum hypothesis naturally led him to check whether certain particular types of sets could be counterexamples. And every type he checked, including perfect sets, was not a counterexample. In fact, that CH is independent of ZFC (as Paul Cohen showed almost 100 years later) means that no set with an explicit characterization can be a counterexample.