Now thats an interesting question.
I think it can never be as simple as an opposition between character and actor.
Truly good actors do characters so well that it becomes difficult to say where the character begins and ends and where the actor begins and ends.
This does not mean that all actors identify with all the characters that they play, i mean one cannot play a 'flesh eating, humanity hating, robot' and identify with all aspects of it.
But for most part, acting is a coming to terms with, an understanding of a particular character... of an entire human life or of a certain point in its destiny... acting, writing, direction are all activities that require deep empathy... an understanding of another mind, another life, another self.
So ideally an actor becomes a character, understands what drives him and doesn't, while not necessarily agreeing with everything the character does or represents...
Of course, in the real world this process may be faster or completely non existent...
For instance, to use my fav eg., the reason i like Gaurav so much is because he became Samay... not just for me but for all adhm fans... and Gaurav himself wrote that he found it difficult at times to distinguish the two, he became Samay so completely... thats why i would love to see Gaurav in a role that he does not identify with at all, to see him push his acting abilities to the fullest!!