I'm currently working on a large scale project doing exactly what you describe. First off, there are so many different ways to do this, so what you really need is some general guidelines to get started, then we can dig a little deeper into details when some initial decisions are made, if you'd like. If you've already got the app deploying and running in two separate steps, but are just looking to combine those, I can definitely help. I'd just need to know how you're currently building/deploying. If you're just getting started on building your pipeline and need to set up the process from scratch, then read on:
First off you'll want to set up some kind of build server that will install your npm dependancies and run your webpack build. Most likely you'll want a separate webpack config that's just for your build server, this'll give you a build optimized for production or qa/staging environments. This config should split out vendor files that you won't update all the time, pull out seperate css files with extract text plugin and uglify the files. If you have an isomorphic React app, or are using es6 features not supported in your version of node, then you'll need a webpack build for your server code as well. This is really different from the hot reloading build you'll want to have on your local machine while you're actually coding the app. I'll be happy to show some examples if you'd like of our webpack config files for both local development and our CI build. You may also need a build.sh or makefile to do something with the compiled .js files that your webpack build creates, but that'll depend on your deployment which I'll cover later. You can run your production build locally as your getting your config just right and fire up the app from those files to test it's all working. Additionally, since you'll likely want to be able to automate all of this, you probably want to run your tests and linting right before you build your app, we run eslint and mocha/jdom to run our enzyme/expect specs as part of our build. Once that's all working nicely, you'll most likely want to set up a build server that can run your builds automatically. My team is using Jenkins for this, which is a little more work to set up, but it's free (aside from the ec2 box we run it on). There are also a ton of subscription based build/continuous integration servers, such as Travis and CodeShip. There's plenty of articles on the pros and cons of these different products and how to set them up. The bottom line is you'll want to have a build server that can pull down your code from source control, install npm deps, lint, test and build your app. If anything fails it should fail your build and if your build succeeds you'll have some sort of archive that you'll later deploy to an ec2 instance. In our shop we use a build.sh file to tarball up our build archive (basically a folder with our node server files as well as our minified client files, css files and any fonts or images needed to run the app) and upload it to an S3 bucket that we deploy from. We like this fairly old school method because the tarball will never change, so we have ultra reliable roll backs.
What you do with your build archive will depend on how you want to do deployments. We have a custom deployment system using puppet, but there are plenty of products that do this such as elastic beanstalk, that would be much easier to set up. You'll want some kind of process supervisor to actually run your node app, so unless you have a dev ops team that wants to build custom pipelines, using AWS built in features will probably be the easiest way to get started. As usual, there are so many ways to do this, but the basic principal is that you need something to download your build archive and run/supervise your node process. You also may want to be able to create and configure ec2 boxes on the fly (Puppet, chef, etc.), or even use containers (Docker) which allow you to move complete stacks around as single units. Using automation to create and configure servers is crucial if you need to scale your app, but it is complicated and may not be necessary for smaller projects. This is definitely an area where you can start simply and add complexity later on, as long as you have good long term goals and make sure to take the necessary step to prepare for future complexity.
All of this can get you pretty far in the weeds, so it's best to find the simplest thing that will serve your needs as you get started and then add complexity as real life situations demand so. I'll be happy to elaborate on any of these details if you provide a little more context about how big and well funded of a project your working on. If it's a little side project to learn the tech, I'd have very different advice then if you're trying to build an app that'll have a lot of traffic and/or complex features.