Again, I've only read the first book, which puts me at a bit of a disadvantage here. The heroine lives in a subjugated and exploited "district" of what used to be the United States, and she's survived some pretty grinding poverty. She hunts illegally to put food on the table, and to provide a meager income for her mother and younger sister.
Her younger sister gets drafted to participate in a nasty public spectacle, something that the ancient Romans would have loved. The heroine volunteers to take her little sister's place, and gets whisked off to the capital and trained to fight. In the games, she has to find food, survive, and be the last kid standing.
The heroine does kill a couple of the other contestants, but she does so indirectly (by dropping a poisonous wasp nest among them, for example), and the people she kills are pretty scary. So unlike Schwarzenegger in The Running Man, she's not snapping necks or wallowing in bloodlust. There are two non-sexual romances (though that could change in latter books), and the author is clearly setting up the heroine to fight the system—to whatever extent is actually possible. But in the first book, the heroine can only make small, symbolic gestures of defiance. She's a teenager in a brutal police state, her family is held hostage to her good behavior, and she needs to play along if she wants to survive. This is a book about trying to be a decent person in a bad society.
Basically, in terms of violence, it's a little closer to something like Terminator 2 than The Running Man—there are several ugly deaths, but the protagonists are trying to do the right thing on some level.
I would have no problem recommending this book to a high-school student. For a 10-year-old? Maybe, but you should definitely read it first, decide for yourself, and be prepared to discuss it with her. It's a really fast read, and I've definitely read worse.
Cheers,
Eric