How Would I Go About Using Node.js For Frontend And Wordpress As The Backend?

To be exact, you can't use Node.js for a front-end solution, since it runs on the server, not the browser (think of it like any other server-side language such as PHP, JSP etc).

You can, however, create the described solution with jQuery or any other Javascript library, you just have to implement data transfer with Socket.IO. On the server-side you'd need something to handle websockets, so the most native way would be to use Node.js, but since you want to use Wordpress, it gets really complicated, as Wordpress is not meant to be used in the way you described, so I'm afraid you'd have to write your CMS from ground up in Node.

Also, the way you described has a huge flaw. Search engine crawlers are still unable to parse and run Javascript, so if all of your content is loaded dynamically, it would seem empty to Google and others, so it would be impossible to ever make it in the search results rendering your site pretty much useless.

For MySQL and other modules for Node, you should check NPM registry and the Node modules page.

EDITAfter Dwayne explained his solution in comments, this is how I'd do it:

1. I'd use jQuery for front-end. Binding the document with .on(), and setting the selector to 'a', so that every anchor on the webpage would fire the handler.
2. The handler parses the a.href attribute and figures out whether it's an external link, which shouldn't be handled by Javascript, or if it's a link to the next page, to an article etc. You can prevent the default action by calling e.preventDefault() in the handler, which prevents the browser from redirecting to the location.
3. Then the handler would get the content in JSON by calling .getJSON() to the URL based on the article. The easiest way would be to have a certain pattern (such as all urls like www.domain.com/api) redirect to the Node service via .htaccess, to prevent cross-domain problems.
4. Node would then see the request, extract the parameters and figure out what the user wants. Then connect to the MySQL database with this module (it's as simple as it can get) and return the corresponding content formatted as JSON. Don't forget to set the Content-Type headers to 'application/json'.
5. jQuery gets the response, figure out the type of the request and updates the content accordingly. Profit.

As you can see, I wouldn't use WebSockets in this case, since you wouldn't really benefit much from it. They are mostly meant for small real-time updates (no huge HTTP headers to reduce the bandwidth) that are both-ways. This means that the server could also push data into the browser, without the browser asking for it. In a blog context, this is not required, and you won't have too many request, so the difference in bandwidth wouldn't be noticeable anyway. If, however, you would like to use it for educational purposes, just basically replace the getJSON part with SocketIO, I'm not sure whether Apache supports proxying WebSockets, though. Extra information about SocketIO basics are here.

Edit: I overlooked the part with 'using Node.js on the front-end'. As Vahur Roosimaa said, Node.js is on the server-side (think of it as Nginx / Apache + PHP combination). Node isn't a frontend library like jQuery. If you want you can use it just for the websockets functionality (I suggest using Socket.IO).

Nice tutorials about Node.js and MySQL:

http://www.giantflyingsaucer.com/blog/?p=2596
http://mclear.co.uk/2011/01/26/very-simple-nodejs-mysql-select-query-example/
http://www.hacksparrow.com/using-mysql-with-node-js.html

This SO question might also help: MySQL with Node.js

Also check the examples from the github repo of node-mysql.

If you want something more advanced like an ORM, I recommend Sequelize.
Another good question from SO: Which ORM should I use for Node.js and MySQL?

You should check out Wordscript which I recently added a Node JS example which can act as a simple front end for doing basic post retrieval from a Wordpress database.

It uses a common mysql library for node, and generates MySQL queries from get parameters and renders data as it is retrieved from the database; including tags.

Wordscript aims to free backend/frontend developers from being forced to work with the Wordpress PHP codebase, but still allows for Wordpress'es administrative interface to be used when needed (and prudent to do so). API's have been written in Ruby and PHP that both return JSON feeds and function generally the same way the node version does; so thats an additional option where a scripting language is available.