Bear in mind that the 5th generation iPod (before it was called "Classic") was known as the "Video iPod", and supported Composite Video Out through the headphone jack using a special, but simple 4-conductor 3.5mm plug that terminated in the standard yellow, red & white RCA connectors. (The 4th generation iPod also had video out the same way, but only for still photos) Several companies made this cable eventually, and you can often find them for $5 or less. Video Out was also available through the dock connector for those who bought the Apple dock, and you could even get better-quality "S-Video" output if the iPod was in the dock.
Starting with the 6th generation, or "Classic", Video Out was ONLY supported through the dock connector, and ONLY by using a special APPLE-made cable, because the cable connector had a proprietary circuit within it. Without the special circuit, the 6th generation iPod would not output any video at all. Kind of a DRM scheme, but in hardware.
Apple made available two different video cables for the 6th gen. iPod Classic (and 3rd generation Nano) -- one for Composite video & audio (yellow, red & white) and one for higher-definition Component video & audio (3 plugs for video - red, green & blue, and 2 plugs for audio - red & white). So if you've got a recent 6th gen. Classic, you need one of those cables, and they're not cheap - expect to pay a minimum of $20 for a 3rd party cable, and Apple's cable is almost $40, but it is also packaged with a power supply brick. (Someone told me you can buy the cable separate from the power supply now, but I haven't checked recently)
Other cable companies eventually licensed the proprietary chip and included it in their dock-connector video cables for the iPod Classic, but beware of cheap knockoffs that don't have that chip - they won't work with the Classic or Nano!