Google Cast Chrome extension still required
Building on its recent expansion of supported apps, Chromecast will now interact with Google Play on the web for Music, Movies and TV shows. Similar to what we've seen happen in the new Youtube player, you'll now find a "Cast" button in the web interface for these two Google Play properties, making it simple to send content from your computer to the Chromecast.
In order for this functionality to work, you'll need to have the "Google Cast" Chrome extension installed to your browser, and then look for the Cast button in Play Music or Play Movies & TV. The option showed up for us after a refresh of the Google Play Music page, and worked right away. Note that the Chromecast won't respond to your computer's system volume, but rather the volume slider in the web page, found next to the Cast button.
It's one more step in making both the Chromecast and the Play Store on the web more powerful, and we love having multiple choices when it comes to pushing content from one screen to another.
Source: +Google Play
