The cable you are describing is rather standard.
Instead of plugging it into your VCR try to see if you have the same connectors in your TV directly. In some TVs the connector of that type might be behind a cover at the front or side of the TV.
You might also need to disconnect you video for this test.
If that works, than your Nintendo 64 is fine and there might be some other issue between your Nintendo 64 and your VCR.