Will being in a different situation than most people of my background affect my chances in any way?

The answer to this is yes-and-no. It probably won't have an effect on how a school will view your stats and grades, aside from the usual way they put that in context with your high school (did you take the most rigorous schedule available, how do your stats compare to other applicants they've gotten from your school—that kind of thing). But if you were to write about it in your essays or even your Additional Info section—being clear about how it's changed the way you see the world and things like that—it could form a really strong cornerstone for your apps and really help your profile stand out.

Going off of what @jcdenton said, I wonder if there are any ways you could take what you've learned from growing up in this situation and apply it to the school? Maybe creating a club, non-profit, or volunteer in some way so you can actually demonstrate how the way you've been raised has made an impact on your life.

Maybe it's a club that helps brainstorm ideas on how to improve the community or educate people on services they can use if they're low income. Maybe you volunteer with elementary age kids who come from the low-income background. Or start a non-profit which provides school supplies or some other critical thing necessary to succeed in school. It's one thing to talk about how growing up in a different situation than your peers affects your view of the world but being able to show that not only has it impacted you in some way AND you choose to try to help out and make a positive impact, I think that would hold a bit more weight on your application.