How do you programmatically fill in a form and 'POST' a web page?

You can use the UploadValues method on WebClient - all it requires is passing a URL and a NameValueCollection. It is the easiest approach that I have found, and the MS documentation has a nice example:
http://msdn.microsoft.com/en-us/library/9w7b4fz7.aspx

Here is a simple version with some error handling:

var webClient = new WebClient();
Debug.Info("PostingForm: " + url);
try
{
     byte [] responseArray = webClient.UploadValues(url, nameValueCollection);
     return new Response(responseArray, (int) HttpStatusCode.OK);
}
catch (WebException e)
{
     var response = (HttpWebResponse)e.Response;
     byte[] responseBytes = IOUtil.StreamToBytes(response.GetResponseStream());
     return new Response(responseBytes, (int) response.StatusCode);
}  

The Response class is a simple wrapper for the response body and status code.

The code will look something like this:

WebRequest req = WebRequest.Create("http://mysite/myform.aspx");
string postData = "item1=11111&item2=22222&Item3=33333";

byte[] send = Encoding.Default.GetBytes(postData);
req.Method = "POST";
req.ContentType = "application/x-www-form-urlencoded";
req.ContentLength = send.Length;

Stream sout = req.GetRequestStream();
sout.Write(send, 0, send.Length);
sout.Flush();
sout.Close();

WebResponse res = req.GetResponse();
StreamReader sr = new StreamReader(res.GetResponseStream());
string returnvalue = sr.ReadToEnd();