The main issue in this case is that none of the available Slingbox models will work with the HDMI cable. Although you can get the passthrough using a cable box or DVR, the HDMI connection will prevent you from getting the video signal.
Besides, it is not possible to get the Slingbox up and running if you hook it up to a TV set. The Slingbox does not have this capability.
The following links will provide you with the information about this situation.