The fuel gauge circuit is one of the simplest in the car. It comprises only the tank sender and the gauge itself. Almost all the problems with the circuit are problems with the sender, so troubleshooting is not complicated. The original units were flimsy and would quit after a time without provocation and would essentially disintegrate if the gasoline became altered or contaminated. Occasionally the fuel gauge itself will fail.
Troubleshooting:
To gain access to the sender, remove the cover panel under the spare tire. The Fuel Sender is the smaller of the two round items and is held in by a black plastic ring.
To test the sender, remove the sender from the tank while leaving it electrically connected (you may need to unplug its 3-wire connector from the harness to get it all the way out). CAUTION fuel will run out of the sender, so lift it slowly allowing it to drain into the tank as you lift it.
With the ignition turned on, turn the sender upside down while an assistant watches for movement on the gauge. You should feel the float drop from end to end as you invert the sender. With the sender inverted the gauge should read full, with the sender out of the tank but upright it should read empty with the low fuel light illuminated.
If the low fuel light does not illuminate, connect the light-green/orange wire in the car side of the harness to ground (the black wire in the same harness will do nicely). With the ignition on the light should illuminate. If it does not, the bulb is likely bad. It is extremely common for the low fuel indicator inside the sender to fail since it is so rarely activated and the contacts get corroded.
As an additional test of the gauge, disconnect the 3-wire connector, and turn the ignition on. The gauge should point straight up. If not, there is an issue with the gauge itself. Check the fuse #5 that powers the instrument cluster. A rather obscure failure is failure of the calibration resistor on the back of the gauge itself. If this happens, the gauge will read about 1/2 normal, i.e. a full tank will read 1/2 full, but empty will still be empty. Replacement of the gauge is the repair.
Service/Replacement:
DMC offers a new Fuel Tank Sender P/N 110555 that fit the original hole and plugs into the original connector.
However, if you are replacing the pump and associated internal tank components (usually due to system contamination/storage failure) you will be better served to replace them with the upgraded Fuel Pump/Sender Module P/N 107000. This replaces all the items that are prone to deterioration - including the sender unit - and is a modern replacement for the original, trouble-prone internal tank components.
Revised DAS 6/22/2016 [edits/references]