Connect your Chromecast and phone to the same wireless network and connect to your Chromecast from any app on the phone that supports Chromecast, like YouTube.
You can download the Chromecast app on your phone for settings or casting your phone screen to your Chromecast.
But basically it's alla about using apps on your phone that supports Chromecast.
Try it in YouTube that you might already have. Press that TV shaped rectangle in the top right corner in YouTube app and select your Chromecast.