I just bought a iPhone 5 & would like to use the airplay function but don't have a receiver or Apple TV to let me do this.

Does anyone know of any alternatives that would let me use airplay so I can get video, photos, music etc to work through my TV & home cinema?

Is there some sort of dongle I can just plug into a spare HDMI slot?

If you have a "connected" or "smart" TV (one that's connected to your router via Ethernet or WiFi) then there are apps such as iMediashare that will allow you to send photos/video/music to your TV.

If your TV isn't networked, then it would work with something like a PS3 as well.