You're looking for the difference in current being fed into the light bulb as a result of increased resistance inherent in a longer wire.

Assume the voltage is constant as that is coming from a steady source. The battery provides the same voltage, but you're reading a lower voltage across the light bulb because there is more of a voltage drop across longer wires and Kirchoff's Voltage Law states that the voltage of the battery is equal to the sum of the voltage drops in a closed loop.

So, given Ohm's Law (Voltage = Current times Resistance), with the same voltage being delivered to circuit, the added resistance provided by the longer wire leads to lowered current flowing through the wire, and thus a dimmer bulb.

If your voltmeter also can measure resistance, connect one lead to each end of the wire when it's not connected to anything. When you have the resistances of each wire you are to use, you can solve for the amount of current going through the light bulb by dividing the voltage by the resistance (V/R)

With higher level mathematics, like derivative calculus, I'm fairly certain that you can calculate the current without having to find the resistance of the wire first, merely by tracking the change in voltage across the light bulb across different lengths of wire, but I've been out of high school so long that I'm afraid I've completely forgotten calculus.

Above is correct, if somewhat complex. Put simply, you need to set your multimeter to measure current (amps) and measure the current draw across the load (bulb) with the different wire lengths. You can also, as noted, calculate this by measuring the resistance on the wire and using Ohm's law.

60 feet of wire will not present a high resistance at low voltages, so you may require a meter that is more sensitive than your average consumer piece purchased at radio shack.