That damn Cloud Word
Keep in mind that a private cloud is really no different than most of your internal web servers and applications today. The article you provide takes it a step further by essentially allowing your dev team to create, scale, and control their own virtual machines. In the end though, with the information you have provided all you are really looking for is IIS to run whatever application you have.
Is There a Difference between Azure and IIS?
The primary different between an Azure web app and one that runs on IIS is pretty simple, one has the Azure application container added to the solution and the other does not. Azure's offering has expanded and you now have more control over the underlying VMs and even your own VMs but I'll approach this from the most basic viewpoint, using Azure as a PaaS or Platform as a Service model.
Visual Studio Configuration
It's pretty easy within Visual Studio to generate multiple build configurations, one specifically for IIS and the other for Azure. This simply defines to build mechanisms that either prepares the application by setting up an Azure deployment package or another which simply creates an IIS package.
Remember Azure is essentially IIS just with configuration and settings files that have to go along with the application.
There really isn't much above and beyond the standard web server setup required here. You may have additional requirements such as file storage and even caching (see Server and Azure AppFabric/Caching). You will have to work with your dev team to determine what the requirements are and setup any necessary systems to support those.
Your dev team also needs to understand something important here:
design and build the application to run in any environment. What I mean by this is that regardless of the final deployment medium and path the application should work normally. So in your situation they most likely will want to use dependency injection for file storage so either the Azure Storage API can be called or it can revert to a standard System.IO file storage mechanism. Either way, just setting up the hardware to support the application isn't enough, it must be designed to support multiple environments as well.