You can continue using WebClient to POST (instead of GET, which is the HTTP verb you're currently using with DownloadString), but I think you'll find it easier to work with the (slightly) lower-level classes WebRequest and WebResponse.
There are two parts to this - the first is to post the login form, the second is recovering the "Set-cookie" header and sending that back to the server as "Cookie" along with your GET request. The server will use this cookie to identify you from now on (assuming it's using cookie-based authentication which I'm fairly confident it is as that page returns a Set-cookie header which includes "PHPSESSID").
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