First thing to check is if the release solenoid is dead. This will require a set of jumper cables and a small screwdriver. Unplug the solenoid connector. Use one wire of the cables to connect the latch or solenoid to ground, like an engine bracket. Connect the other wire to battery positive with one clamp, and to the screwdriver shaft with the other clamp. Touch the screwdriver tip to the terminal inside the solenoid connector, this should cause the solenoid to fire. If it does, use another scredriver to close the latch and try again. If it releases ok, then the next thing to check is the switch.
The most common problem with the switch is that someone pushes down on it because the little trunk symbol is on the bottom. This snaps off the indexing key inside the switch, rendering it junk. The easy test for this is to gently push down on the switch and see if it goes all the way down with no resistance. Normally it will not go down at all, whereas it will go up with minimal resistance. If the switch is bad you'll need to get another one.
Next is a bit more complicated. There are switches in the park brake handle, the ignition switch, and I think in the auto tranny shifter linkage that all act as safety interlocks for the trunk. What makes it hard is that every year Pontiac changed around how the interlocks work and I never can remember them all. You will need a schematic for your year car to trouble shoot the wiring, but first you should check the easy and common problems as listed above.