Daenerys would be hard character for anyone to play, much less a 25-year-old with only a couple of other credits on her resume, one of them a Broadway show that is about to close. Perhaps this is simply a near-perfect match between a performer and the one quality that must be conveyed for this storyline to work. But it's a tough one, and she's doing a great job.
Dany is utterly single-minded, and how to play that without turning people off? Her only goal in life is to take back the notably ugly throne she regards as a family heirloom. The one thing we need to believe above all, in order to become invested in that quest, to not regard it as delusional or psychopathic, is that she really could be a queen -- even more, that she deserves to be one. Quite a feat for one so young.