You can't connect the Apple TV to the cable box either using HDMI or something else. The Apple TV is for use with HDMI only, although some users have found workarounds which may or may not work for you.
Some users with DVI have managed to get their TV's to work with DVI-HDMI cables. DVI carries no audio, so alternative connections need to be explored to enable audio. DVI doesn't necessarily support HDCP as well as other standards used by HDMI (which may or may not be an issue)
There are hardware converters that will convert HDMI to various other types of output, however there are some issues with doing so that you should be aware of.
Resolution and aspect ratio
I'm not aware of any converters that will scale the output from the Apple TV, any TV or projector which is used will need to be widescreen and support resolutions of 720p (Apple TV 2), 720p/1080p (Apple TV 3)