I'd like to start developing Google Chrome extension using node.js (since I've already written a "text-to-song" script in node.js, and I'd like to turn it into a Chrome extension.) What would be the most straightforward way of approaching this problem?
Actually it is. Look at this Developers Live-cast. This is something I've been looking for as well, and this would help you.
This brings your node applications bundled to your browser. Here is the repo!
I've noticed that this old answer of mine keeps getting upvotes now and then (thank you all).
But nowadays I'm more an advocate of using web apps instead of bundling your application into many platforms like the chrome store or whatever. You can check the google's post here and here indicating some directions.
In practice I advise for you to start building a progressive web app (PWA) with offline capabilities using service worker and progressive stuff.
There are plenty of resources around the web nowadays and you can offer a much richer application that may achieve a much broader audience if you do it the right way.
Thanks again, and good coding.
Simple answer is NO, unless you can find a way to install node.js with an extension using NPAPI.
Google Chrome Extension
"Google Chrome Extensions are small software programs that can modify and enhance the functionality of the Chrome browser".
If you wish for others to download your extension you will have to provide config.json file that describes you extension sets permissions etc.
If wish for others to use you application you start you nodejs server and listen for incoming requests.
Despite some of the similarities a Google Chrome Extension and Nodejs have nothing to with each other. You cannot use them together in some special way outside of the normal client/server communication.