I"ve noticed with many beginners that the C major chord seems to be a toughie. I must have been lucky because it was always the Dmajor that got me in the beginning. Anyway, from what I've seen, new people tend to not have their wrist as "upright" and it makes those middle fingers touch other strings. I'm not sure if thats your problem as I can't see your fingering. Try bring your wrist up a few more degrees and maybe it'll help.

Persistence is indeed key, as well. I thought I'd never get those chord changes way back when.