You can't connect Apple TV to the directv receiver. It connects to your TV and requires an HDTV and an HDMI port. There is no guarantee anything else will work. Furthermore, you wouldn't be able to watch HDCP content (i.e. iTunes, Netflix) and would have issues with resolution/aspect ratio.
youre right! I tried buying a DVI to HDMI adapter and then using audio cables with apple tv on a 2004 Sony Wega Trinitron that is still working and for a heavy TV works well, but is not true HDMI. Therein lies the problem, like andrew says with aspect ratio. You have to get an HDMI TV for apple tv to work correctly. No back door attempts will work.