As you say, a grounding problem.
Both the bare AND the black wires (assuming you got 4 conductor) need to be soldered to a ground point (usually the back of the volume pot, which is in turn grounded to the trem claw).
A few questions:
Does the noise change when you touch the strings, bridge or tuning machines?
Does it only happen when the pickup selector is switched to the MM (assuming you only got the bridge)?