I think Emilia Clarke over acts a bit and I find her a bit out of class when compared to the rest of the high quality cast. She narrows her eyes to show her wisdom and courage. That's really all she does. I think that is one role that could have been casted better. Her and Caitlyn Tully.
But, I find the story of her quest to be a bit boring anyway as it tends to just lead to her squinting her eyes and then bringing out the dragon. The battles of the other houses are far more unpredictable.
I think that what you're describing is more under- than overacting. Regardless, her story is what the writers give her, and it's difficult to imagine Dany's dragons not taking a leading role in her conquests. It's also in line with the history of Valyria, where dragons apparently did the same thing. And, to be fair, her elimination of the Dothraki khals was her own doing, with some help from Daario and Jorah.