You just need to purchase a subscription from any hosting provider. There are multiple solution providers that provide competing packages for the companies and individuals. You can select from one of them. Since this is ASP.NET, I'd like to say you should go and visit here,
ASP.NET Hosting | The ASP.NET Site[
^]. Start by having a look at Microsoft's cloud platform. There are other third-party options also provided along with.
I also wrote an article that explains how to do this on your own machines,
A simple guide to setting up home server using IIS and ASP.NET[
^]. I used a local network for hosting the web application, you can use a static IP address that can map to Internet.
Also as already said in the comments, you just need to host the web application. Console applications are the clients for your web API that would consume the API through proper HTTP requests. You don't need to worry about that. You just have to publish the application. Read that article that I have shared, it would give you an idea how to do that. If you are going to get a third-party solution, chances are that they will guide you through set up also.