Having only seen a handful of the first season of Justice League, I can only comment on what I've seen of her. Since I liked Hawkgirl in the comics, I really have to say I wanted to like her in this but from what I have seen, this failed to give her any dimension really. I did find it odd that they included her without Hawkman, but I could see in a way why they didn't. It might have made her seem more like a sidekick than a major character. Still, in the poll, I clicked on liking her because I still think like the fact that the creators chose to use her since she was virtually a rather unknown character for the most part before this.
However, I do have a question about her, while speaking of her. In the episode War World, she and Green Lantern are being transported supposedly to their destination to find Superman and Jon in some alien's spaceship. In close quarters, there isn't much room and at one point, Shayera says "Ow, would you mind getting off my wing?" Since when were her wings actually a part of her body, with nerves and all? In every comic I've ever seen, the wings were part of the costume, so what's the deal there?
I don't want to spoil anything for you, but those questions are answered. Hawkgirl got very little development in season 1, but starting in season 2 she gets a lot of focus. You really must watch the future seasons of JL and JLU.