I would like to stream my PC games from my computer to my Roku 3. I can use my wireless Xbox 360 controller to play. Is this even possible, and if so how can I do it?
You can't, not in realtime. A streaming networked device like the Roku will always introduce a large delay, at least in the high hundreds of milliseconds, but probably multiple seconds. Your screen and audio would feel very out of sync.
To get a realtime audio signal, you have to have a very fast connection. With today's networking technology, that means that a networked device is out of the question and you need something that connects without using networking protocols. A direct audio cable, or a wireless (non-networking) audio system would be needed to get the speeds necessary to avoid audible lag.