I find it much easier to just use one of these. This splits the signal for you straight out of the console, so you can plug S-video into the capture card and the composite end into the TV. You can then split the audio if you want, or put one audio channel into the TV and one in the card (in which case you'd have to make sure the audio is mixed to both channels), or both into the card (in which case you'd have to listen through your capture program). I do the latter.
I can give you some sample videos of what this looks like, recorded with Amarec/LAGS codec. For such a simple set of equipment you get about as good of quality as you're going to get out of a normal N64.
S-Video is when the luma (luminance) and chroma (chrominance/color) signals are separated rather than combined.
https://en.wikipedia.org/wiki/S-Video
https://www.amazon.com/S-Video-Cable-GAMECUBE-Super-Nintendo-64/dp/B00HRJWCBC
https://www.nayuki.io/page/nintendo-64-over-svideo
If you still can't get any video output, then either:
P.S.
Can you get a picture of the other side of the board?
If it's this one, I'd be wary of it. A lot of the time the ones with both composite and S-Video connections are poorly-made and will result in really lousy S-Video quality. At least one reviewer noted diagonal interference, which a well-built cable shouldn't have. But if it looks good to you, then that's what matters.
If you want something really high-end, Retro Access makes an S-Video cable with real 75 ohm coaxial cabling, though they're handmade so you'll have to wait until they're taking new orders. If you want something cheaper and immediately available, I've heard good things about these ones.