I have a Mid 2010 MacBook Pro with El Capitan, I'm not able to use AirPlay because I simply don't have the hardware for it. However, I'm trying to work through the thought experiment of how I might be able to connect my computer to an Apple TV. Is there some external hardware that I could hook up my computer to the Apple TV (Gen 3) through Airplay (the same way that I could use an external wifi usb)?
There is no external hardware to use airplay, if your aim is specifically to connect your MBP to AppleTV through AirPlay I think you are out of luck. There may be very tricky ways of doing this, although much of the encoding used in AirPlay is done by hardware(I believe Intel chip) that your laptop doesn't have (as far as I know).
There are other options though for wirelessly connecting your laptop to a screen, but thats out of scope I guess.