I have built and tested a RShiny dashboard locally on my computer. I would like to host this dashboard on an AWS EC2 instance. I am comfortable spinning up the instance and getting the files transferred. My concern is how to restrict access to this dashboard to users at my organization. In addition I would like to have a more informative url (I assume this means digging into DNS, but not sure how to go about it).
Since your post doesn't provide much detail regarding any existing infrastructure you may (or may not have) in AWS, I'll assume none and give you the high-level approach I'd take to host your app net-new:
This is a high-level playbook only. There are tons of details you'll need to work out as you go through the process (e.g. VPC network design [CIDRs, subnet sizes, # of availability zones used, etc.], how you manage the EC2 [create an AMI, use an empty AMI and just SSH your files up, git-clone the files onto the instance, etc.]).
Additionally, this post doesn't address more advanced techniques of resource management such as infra-as-code approaches (e.g. Cloudformation, Terraform, etc.), but this is something you should consider.
Lastly, while your question mostly focused on AWS, please do consider the deeper aspects of AuthN/AuthZ within your application itself, particularly if this dashboard is hosting sensitive data. Also, consider any administrative interfaces within your shiny application and make sure the AuthN/AuthZ solution offers adequate protection of these interfaces such that your other users won't be able to access them.
A few other closing thoughts... There are other ways to do this. For instance, you could:
In my opinion, if you are just getting into networking into AWS, the VPN solutions will be messy and more complicated than the setup I detailed above, but I did want to address them for the sake of completeness. The "tried-and-true" approach of a load balancer will work for your use-case, albeit this will be a bit complicated if you don't have any familiarity with the underlying networking concepts.