I believe that eyeTV is only an input device, so you won't be able to do that with it.
Depending on which Mac you have, it could be as easy as an AV cable.
Most Mac portables have some sort of AV output. Either composite or s-video.
The new iMacs have it and I think the eMacs as well.
What Mac do you have?
EDIT - I just looked at your signature. The PowerBook that you have will definately be able to output to a TV via the s-video port or you could use the included s-video to composite adapter. You'll also need a mini-jack to RCA adapter to output the audio.