I checked the MSDN on DocumentDB for .Net (here) and found 3 valid constructors. However none of them makes use of connection strings, which sounds strange for me.
I
That works for me:
private static DocumentClient InitializeDocumentClient()
{
string connectionString = ConfigurationManager.AppSettings["CosmosTest"];
string[] connectionStringParts = connectionString.Split(';');
Uri clientUrl = new Uri(connectionStringParts[0].Split('=')[1]);
int keyStartPosition = connectionStringParts[1].IndexOf('=') + 1;
string clientKey = connectionStringParts[1].Substring(keyStartPosition, connectionStringParts[1].Length-keyStartPosition);
return new DocumentClient(clientUrl, clientKey, connectionPolicy);
}