As of 2015, the main phone number for Chromecast technical and billing support in the United States is 855-836-3987. International support numbers are available on the Chromecast Help page on Google.com.
Chromecast recommends that customers reboot their routers and their Chromecast devices before calling in for support. To do this, unplug the Chromecast device from the power source. Wait 30 seconds, and plug the device back in. Next, unplug the wireless router from the power supply, and wait at least 30 seconds before plugging it back in. If the problem isn't resolved after the router reboots, check the Chromecast support page for simple troubleshooting tips.