I recently purchased an Amazon Echo Dot, and set it up to work with my Chromecast thanks to this Github Project. I dove into the inner workings of the project to understand the underlying communication between the different components, and simply put, the local server component just sends my Chromecast a URL to play based on what is sent down from the Lambda function. It's actually pretty straight forward after you do some digging (which I had to because the YouTube wasn't working out of the box). Knowing how all this works, it should be pretty easy to send my local server component a URL for a Plex stream and have it play on my Chromecast.
That being said, I don't want to build my own Alexa Plex Skill and this one is no longer being supported (and missing a few key features). I'd like to utilize the source code of the official Alexa Plex Skill and modify it to publish the media's URL to the Amazon SNS that my local server is listening on rather than send a command to a Plex client (since Chromecast is not a Plex client). This way, my Plex media will play successfully on my Chromecast through Alexa and all will be right with the world.