Spotify famously said it has no plans to support Chromecast, but that hasn't stopped one Android developer from coming up with a solution. Spoticast is the work of NOP Developments and while it's not perfect – yet – it provides Spotify Premium subscribers with at least one method to get their music onto their TV.
To make it work at all you have to enable "Device Broadcast Status" in Spotify and restart first before anything will work. There are a couple of issues present, one of which is that your device will continue to play the sound and there are some delays and interruptions. But it's a start, and let's not forget it's a workaround, too. What we really want is proper Spotify support. Except we're not likely to get that.
Spoticast is free to download so grab it from the Play Store link up top and give it a try.