A cast button should appear in the toolbar of the app whenever a Chromecast is detected on the network by Google Play services.
If this doesn’t happen, the first thing to do is to ensure your device is capable of casting. Start the Youtube app and ensure you can cast from it. If you cannot, ensure you have the latest Google Play Services installed on your device. This troubleshooting page from Google may also help. If all else fails, attempt to reboot the mobile device and Chromecast.