How can i befriend a girl who seems to be having trouble?

there's this girl in my school, and i feel like she either might not want to be there, is depressed, doesn't sleep enough, or there might be something else wrong. granted, this may just be her personality, but i feel bad and want to become more friendly towards her. how can i act more friendly towards her without being overbearing or annoying (i don't want her to think i'm that annoying smart girl who talks with everybody and won't leave her alone)?