Let's consider a practical example by saying that you want to check whether the port for eMule is being forwarded appropriately. eMule needs to be able to accept connections from other computers on the internet. When that happens, a remote computer tries to connect to a specific port on your IP address and goes through your router. The router in turn is being configured to know that if some external computer tries to connect to it on that specific port, then it should pass that connection to another computer that resides in your internal network.

So, in order to properly check if a port is being forwarded correctly, you would need some program that resides outside your internal network to try to connect to that port. In order to do that you need two things:

1. One program on the computer where the port is being forwarded to (internal network) that is listening on that port.
2. One program on the internet that tries to connect to (1).

That is why I am saying that you will most probably need to deploy program #2 somewhere on an ASP.Net host - so that the program is running at the internet and can check for open ports (and therefore correct setup of port forwarding). If that's not clear, try reading this article.

As to where you can start, as I said you need to have a look at the System.Net.Sockets namespace. You may google for code samples (such as this) but I would also recommend that you take the time to familiarize yourself with this particular namespace.