Step 0 - Install Chrome/Chromium
Streaming to the Chromecast from your PC currently requires a Chrome based browser. Install either Google Chrome or Chromium.
Step 1 - Install Google Cast and TCast Extensions
Select each of the addon links below and add them to your browser:
Step 2 - Enable Streaming from your Channel of Choice
Open the twitch.tv stream URL you want to cast to your Chromecast. Play your stream and then click the Chromecast button underneath the video player. You will then see the following appear over your stream:
Next, click the Chromecast icon in the upper right hand corner of your browser and select the Chromecast device you want to stream to:
Your Chromecast should then take a minute to load up the stream and then it should soon be playing! In your browser page you will then see the following screen from which you can adjust the stream quality and play/pause the stream:
And you are all set - enjoy streaming twitch on your Chromecast device! Have any questions or issues please feel free to leave a comment below and I'll do my best to help.
I've only tested the above instructions on my Linux based PC, but these extensions should work on any platform that supports Chrome/Chromium.