For the sake of the rhythmic analysis I'm doing, I had the idea to solder a wire to a metal pick, ground the strings and rig something up that sends data to a computer every time the pick touches a string and completes the circuit. But would doing so affect the sound of the guitar and be heard once amplified?
Time for some physics!
A guitar pickup consists of two parts: a permanent magnet and a coil of wire. These can sense the motion of nearby ferric object (like a steel guitar string) by taking advantage of Faraday's Law, which says that whenever the magnetic field passing through a loop or coil of wire changes, it will cause a current to flow in that loop. In the case of a guitar pickup, the permanent magnet causes the steel in the string to become magnetized, creating an extra magnetic field through the coil. When the string is plucked, it moves closer and farther from the pickup, causing the magnetic field it sends through the pickup coil to change. This then causes a current to flow through the coil, and the amount of current in the coil fluctuates at the same frequency as the string. This current is the signal that is then multiplied by the amplifier.
If you were to run a current down a steel guitar string near a pickup, this current would also create its own magnetic field, and some of this magnetic field would pass through the pickup coil as well. This means that the pickup could in principle sense the presence of the current; but because of Faraday's law, the response of the pickup would depend on the rate of change of the current rather than the amount of current. A straightforward direct current would create a magnetic field that wouldn't change with respect to time, and so it wouldn't cause any additional current to flow in the pickup coil. On the other hand, any change in the amount of current flowing down the string would cause a changing magnetic field, creating an additional unwanted current in the pickup coil.
You mention that you will be connecting this to a computer, so you're dealing with DC currents while they're flowing. You'll thus have a constant amount of current flowing through the string, changing only when the pick makes or breaks contact with the string. I suspect that if your proposed setup has an audible effect, it will be in the form of a "click" or "pop" at these moments, since at that point the amount of current flowing through them (and the magnetic field they produce) is changing quite rapidly.
I would be very curious to know what the result of your experiment is, and whether my prediction is at all correct. Please do report back if you can (and don't electrocute yourself in the process!)
It would not do all that much since the magnetic field lines of any current would form circles around the string and thus be perpendicular to what the pickups will pick up. Also there is no necessity for using significant amounts of current here.