The way we do this is connect a Line-Out from our VCR or other analog sound device into the Line-In on our sound board (either card or circuitry built-in to your motherboard). You'll need some sort of software that expects to listen to this input and that will save it to a usable file-type. I doubt you'll find software to use your TV card, but I may be wrong. No, the headphone jack is not likely to work very well with the line-in. You'll probably get some sound, but it will likely lack robustness and clarity.
You'll probably need to dig up another source to output the audio, but if you don't have usable Line-in for your computer, your camcorder may have one. In that case you could merely pass-through via the firewire cable, into your NLE (audio only).