If running an ASP.NET Web Site outside of IIS is your requirement, you picked the wrong architecture.
The only real way this will work is if you create a normal windows service with all your code in and that code also generates static html for your browser to parse. But that's all you can really do.
Browser connects on port and sends the normal request to a listening socket in your service. Service gets that request and outputs the HTML that your code generates, based on the command it received.
You're effectively writing a web-server, which in itself is an interesting project, but wholly unnecessary.
There's not that much wrong with IIS, why can't you use it?
PS. This is what a typical browser request looks like
So you would need to parse that on your socket to make a decision about which method to run in your service.
It's possible, but it's bad. VERY BAD. Use IIS, or if you're running Linux, use the ASP .NET plugin for Apache. It's based on Mono so isn't completely up to date with the latest release but most major functionality is there.
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