If I place a compass over a wire(such that the wire is positioned north-south) with charge flowing through it, and it points northeast, how can I determine the direction of the electron current flowing through the wire?
Here you can see the magnetic field. This answer your question right?
A mnemonic memory aid to help you understand it is the right hand rule.
Stick your thumb out on your right hand, and imagine holding the wire with your thumb pointing in the direction that current is flowing. As your fingers curl around the wire, so are the magnetic field lines circling the conductor. Think of the base of your fingers as starting at north and wrap around the conductor to the south at the tips of your fingers. In other words the magnetic field is always at 90 degrees to the current flow, and with the right hand rule tells you which direction the magnetic field lines flow.
A compass will be most affected by the magnetic field of a current carrying wire when the plane of the compass is at 90 degrees to the wire, and lest effected when the plane of the compass is in the same plane as the wire. When a compass is placed on one side of the wire, it will point in one direction. When placed on the other side it will point in the other direction because the compass was moved 180 degrees around the wire and so has the magnetic field rotated by 180 degrees.