I just got a Chromecast for Christmas, and either I'm doing something wrong, OR you need an iOS/Droid device to stream to the Chromecast in order to watch Netflix on TV.
Download the Netflix app from the App Store....If your phone is on the same Wireless channel as the Chromecast, you should see the Chromecast logo in the upper right part of the Netflix screen. Click on the logo, and it should list all of the Chromecasts avaiable, then select the one you want and you should see it on your TV.