I didn't think the NES used RCA cables, but then again, it's been forever since I had one (

) so I could be wrong. If it's got the gray box with a co-axial threaded knob on it, then I believe you run the co-axial that goes from your TV to the VCR into the NES instead. Put it on Channel 3, and I think it would work. You may have to mess with VCR settings if you run it through the VCR.
If that doesn't work, clean out the system using Q-tips. Get all the dust out. Same with the game. Blowing on it really hard (ok no jokes please

) will help clean it out.
Make sure you're on Channel 3 (or 4 if the NES is set to it...But I don't think they started making systems with channel selectors until SNES), and make sure all the wires are connected tight.
From the problem you've mentioned though, I'd just say the system or game is dirty, or the game is broken.
-Jin