How do I play my music library on a Chromecast TV?
Casting to a Chromecast generally does not involve your device streaming to the Chromecast - you need to run some sort of server to stream from a local computer (or the Chrome extension that mirrors a computer screen to the Chromecast.) 

When you use apps on iOS to cast, it just tells the Chromecast where on the internet to stream from - and the internet service must support Chromecast.

So what you want to do with your music app can’t be done. Depending on where you keep your music there may be some solutions. For example, if you have a computer with your music on it, you could install Plex as a server, and the Plex app on your iOS device.