public class CookieWebClient : WebClient
{
  public CookieContainer CookieContainer { get; }

  public CookieWebClient()
  {
    CookieContainer = new CookieContainer();
  }

  public CookieWebClient(CookieContainer cookieContainer)
  {
    CookieContainer = cookieContainer;
  }

  protected override WebRequest GetWebRequest(Uri address)
  {
    if (!(base.GetWebRequest(address) is HttpWebRequest request)) return base.GetWebRequest(address);
    request.CookieContainer = CookieContainer;
    return request;
  }
}
...
var client = new CookieWebClient();
var loginData = new NameValueCollection
{
  { "login", "hacker_vasya" },
  { "password", "12345" }
};
client.UploadValues("http://www.sql.ru/forum/login.aspx", "POST", loginData);