You'll need a web site, or at least some other service on the network that will talk to the database. (You don't want anyone connecting directly to the server, as any program that does will need to have the password -- and someone inclined to mess with your data could get that password in the assembly with very little trouble.)
On the server, it depends what language/platform you use. For web / MySQL, i'd recommend PHP, as it come semi built with support. And frankly, it's easier to start with than ASP.net is.
Once you have something to handle that access, in your C# app, you can use a WebRequest to post the info and see the response. Something like this:
WebRequest req = WebRequest.Create("http://my.site.com/reportscore");
req.Method = "POST";
req.ContentType = "application/x-www-form-urlencoded"; // or something like that
Stream reqStream = req.GetRequestStream();
byte[] encoded = Encoding.UTF8.GetBytes("score=12345&name=Somebody");
reqStream.Write(encoded, 0, encoded.Length);
reqStream.Close();
WebResponse resp = req.GetResponse();
The response object will have a stream you can use to read the data the server sent back.
Something to keep in mind, though...anyone that knows this URL, and what to send, could mess with your scores and give themselves a score of 24246246624245245245 if you don't have some way to validate the score.