Q: I have a brand new iPhone 5c and I cannot find airplay; does this model not have airplay?

It should - Airplay mirroring that is.

Restart all devices including your router and check Airplay is actually enabled in AppleTV menus.

Do you actually have an AppleTV?  Without one, AirPlay would not display as there is nothing to send AirPlay content to.

Are both devices connected to the same network?  This is a requirement for AirPlay to function.

If they truly are, then there is a network issue causing the problem.