You're close. Convert to base64.
(The other answers don't directly relate to your need for a WebRequest.)
public static string GetAvatar()
{
HttpWebRequest request = (HttpWebRequest)WebRequest.Create("http://api.mxit.com/user/public/avatar/vini.katyal");
WebResponse response = request.GetResponse();
Stream responseStream = response.GetResponseStream();
MemoryStream ms = new MemoryStream();
responseStream.CopyTo(ms);
byte[] buffer = ms.ToArray();
string result = Convert.ToBase64String(buffer);
response.Close();
responseStream.Close();
return result;
}