The car backfires at ignition, through the throttle body, and it's my opinion the distributor wasn't installed correctly. How do I correctly install a distributor.?

It takes Fire, Fuel, and Air along with the correct ignition timing and decent compression for any engine to run.

A coil requires at least 9.5 volts when cranking, for the plugs to receive enough spark to ignite the fuel so make sure your battery is fully charged and that you're getting a hot spark.

Verify that the engine timing is correct. #1 cylinder at TDC on the compression stroke, both valves closed, and, the ignition rotor pointing at the plug wire contact for cylinder #1. (sometimes when installing the distributor it moves too far out of position and is not aligned with where it should be, you have to adjust when installing it to make sure it points to #1 within the adjustment range when installed)

Spray some carb cleaner into the intake and see if it starts for a minute. If it does, it's a fuel problem. If it backfires, it's a timing problem.

Rick may have placed his question in the wrong category but I'll give it a shot... First, Rick, I'm going to have to presume the 'carb' is actually a throttle body and the wires are going to the injector spray nozzles on the top of the 'carb' throttle body.
If your engine is backfiring through the throttle body it can mean one of three things,
!.) First if it runs, or even won't run, double check the spark plug wires to make sure #1 wire goes to the #1 cylinder and the rest are to the correct cylinders too. Just two wires crossed will backfire through the intake with flames. For referencing which is where, The #1 cylinder on 350 small block Chevy is the forward most cylinder on the drivers side of the engine. The cylinder behind #1 is the #3 cylinder the #5 and #7 on the drivers side. The #2 cylinder is the forward most cylinder on the passenger side with #4 behind it then #6 and #8 in the rear. The firing order for all Chevy V-8's is as follows; 1-8-4-3-6-5-7-2. Looking at the engine from the front of the car the #1 cylinder tower on the distributor cap is usually at the 5 o'clock position or pretty close to it. The distributor rotates in a clockwise direction.
2.) The second condition could easily be the distributor timing is off. If you haven't turned the distributor any then that isn't the problem. If it runs double check the ignition timing for accuracy.It should read between 8 - 12 degrees advanced.
3.) Lastly, and this may be difficult to swallow if it's true but here goes. You could have either a sticking intake valve or a flat cam lobe on one of your cylinders. To check this pull the wire off the distributor cap and put it back on one by one to see if the engine runs differently with it off. If no change if found that is your failing cylinder to investigate. By that I mean to check the compression reading on that cylinder.
Changing Distributor Cap and Rotor Chevrolet Truck GMC 350 305 How to 1988 95 GM Truck Ignition Systems

100 psi per cylinder is way too low--should be in the vicinity of 150-165 per cylinder
back firing indicates wrong firing order and as you have replaced the plugs etc check that you have the leads on the correct plugs in the correct order
the timing for the engine should be around the 10 degree btdc and not tdc

Have you checked your timimg? Sounds like it is set to fast

Did you also change the Coolant Temperature Sensor? If it is reading wrong, the computer may not realize the engine is cold, and won't increase the gas for that.
Spray some carb cleaner around the outside of the throttle shaft, if engine speed increases, that could be a vacuum leak. I've never heard of that, but anything is possible. It sounds like it would have to be a serious air or vacuum leak to cause your problem. Have you blocked off the power brake booster to check if that is leaking?

You might want to do a compression test to see if there is any internal mechanical damage.

Sounds like a condensor or breaker point problem. When it just starts backfiring while running it is certainly ignition related. Most of the time condensors are the culprit but it can be an ignition module or distributor breaker points / sensor (if so installed.) Should not be to hard to sort out. I know most Vette owners are long ago or soon to be mechanics :)

Thanks for using FixYa!

Kelly

It sounds like you may have an intake valve stuck open. the main cause of this is carbon build up. then once you clean it it breaks free and a chunk sticks in the intake valves. other causes of this and check these first they are easier, are crossed ignition wires (ensure you have the correct firing order) Bad ignition wires (allowing the spark to "jump") water or condensation in the dist. cap. or incorrect ignition timing. and dont rule out a possible ECM failure.

Check your ignition wires, rotor and cap. Backfiring is caused by ignition misfire or worst case, a burned intake valve. Your distributor is locked in place so timing should not be an issue but the pickup inside the distributor may have become loose and then can cause this as well (but happens very rarely)

Ignition timing is set by the computer as long as the distributor pickup point is within reason.

It could be running too lean. One of the TBI injectors may be sticking or not working (wiring or computer problem), or fuel pressure could be low.

first check to be sure the vavuum lines at the throttle body are in there correct places.
on the main pulley of your engine, the one on the crank shaft there will be a little notch on it. mark the notch with some chalk or a dab of white paint. then loosen the bolt that holds down the distributor until you can turn the distributor. turn the distributor a little bit, snug the bolt up and try it. once you have the car running you will need a timing lite to set the timing to the correct timing mark. which should be 8 degrees before top dead center for you car.