unless there is additional information that you need to save along with the image there is no need for serializing it .. with that said if you happen to desperately serialize it, first read it in a byte array and use the byte array in your serializable class.
Image img = Clipboard.GetImage();
MemoryStream ms = new MemoryStream();
img.Save(ms, System.Drawing.Imaging.ImageFormat.Jpeg);
byte[] imgByteArray= ms.ToArray();
hope this helps
good luck