Apple doesn't make it easy exactly. This is because they would prefer you buy an Apple TV rather than a Google Audio Chromecast. As you might expect, the Apple TV is very nicely integrated with all Apple products and, for some, the added expense may be worth the added cost.
That said, if you want to play your iTunes music using your Chromecast I would recommend that you download the iOS Google "Play Music" app, sign up for a Google account if you don't have one already, then upload your iTunes library to your Google account. If you use Chrome, they have a nice plug-in application called "Google Play Music" that automates this for you. In other words, it recognizes you have an iTunes library and then with a single button push it will do all the transferring. Alternatively, you can direct the tool towards a folder on your Mac and upload tunes that way. I understand this is a bit of a hassle but once uploaded, you can use the "Play Music" app to play your songs.