2 years ago I tried to set up a streaming server using Adobe Flash Media Server and Adobe Media Encoder. At that time they were still free IIRC. I used a virtual camera software to capture gaming footage and encoded it with H.264 codec using the Media Encoder. Then the stream was sent up to the adobe Flash Media Server run on a remote Windows server. By embedding a flash player on my website, visitors are able to see the live stream.
But now Adobe Flash Server isn't free any more. I'm wondering whether there are equivalent video streaming solutions that are free to use.
I think FFMpeg and FFServer are the best solutions for you.
It's free, powerful and open-source project.