Figuring out a small electrical problem

I've got an exterior twin spot light fixture that's gone dark and I'm not sure if it's the fixture itself or the electrical line leading to it. Everything is so old it could be anything, honestly. The fixture is dark activated, so I changed out the round sensor with a new one from HD and it still doesn't light up. Actually, the bulb in each side lights up just slightly, so I thought it was the sensor, but was wrong.

How do I test the wires leading into the fixture? I've got a multi-meter but could use some pointers on using it for this issue. FYI, there is no ground and the 2 Romex wires aren't guaranteed to be hooked up correctly and the right color.