AirPlay Mirroring, the tech that allows you to display your iOS device's screen on your Apple TV, it great, but you need an Apple TV to use it. Instructables user edyboy12 shows how to use a Raspberry Pi instead.
You'll basically only need to download and install some software for this to work. The Instructables guide uses software called rPlay, which is still in beta and includes a random licence key, but it seems to do the job. Head over to Instructables for the full guide to doing it for yourself.
AirPlay Mirroring without Apple TV [Instructables]