Hmm. That's a tough one.
You could have an internal alternator fault. You might try disconnecting it, to see if that helps.
To check your ground, you can take a small wire and attach it to the negative battery terminal (not the cable, but the actual terminal itself), with both cables hooked up to the battery. Then connect that small wire to the body and the engine. Then start using electrical stuff, like the starter, head lights, etc. If the small wire gets warm or hot, or the electrical components work properly, then you definitely have a ground issue. I would also grab a volt meter, and start probing stuff. If you put the negative probe on the negative battery terminal, and probe the brake light or headlights (when they are tuned on), are you getting 12+? If so, is it constant or does the voltage wander around? You might also look at any fuseable links and any main power relays. You can swap relays around to test them.
Wiring is not much fun....