WARNING: Post contains Season One spoilers!
Claire in Canon
Claire Bennet is just a cheerleader from Texas. Or at least that's what she'd like to be. Instead, she has the extraordinary ability to regenerate. When she gets hurt, she heals. It doesn't matter how it happens. She can fall from many stories high, get her hand caught in a garbage disposal, cut herself, get shot, or even be exposed to radiation. It doesn't matter. She will heal as though nothing had happened. The only time she "dies" is if something is stuck in her brain stem, like a tree branch.
Claire was adopted by the Bennets when she was just a baby. Mr. Bennet was only meant to keep an eye on her. He was not meant to love her as any father would love a child. Therefore, he has done all he could to protect her. He has protected her from an agency interested in those with abilities. He has also protected her from Sylar, a psycho who had super power envy. He was not born with any abilities, but he has been able to pick up abilities by eviscerating the brains of those who do.
I am pulling Claire from after the season one finale. From her point of view, things are safe now. Sylar is dead, or so she thinks. The bomb meant to destroy New York City has been averted. Unfortunately, it looked as though her biological father Nathan and his brother Peter were killed saving the city.
Claire at Milieux
What does this mean for Claire at Milieux? She believes they perished, even though Peter has the ability to regenerate thanks to absorbing it from Claire. She does not believe he'd be able to survive being blown up. So she has issues with losing people she didn't get a chance to know well. Thelma can now be added to this list. Part of her hopes Peter and Nathan (who has the ability to fly at super speed) somehow survived.
She is also keeping all of this a secret for now. She desperately wants to be normal, despite living in this town.