1988 Plymouth Reliant shakes

Is it shaking when your just idling or when your driving at highway speeds? If it shakes when in Park before you even get it going you may have a miss in one cylinder. Does it ever go away?

It might be missing a cylinder under load and high vacuum conditions. If you remove and check the plugs they might be fouled and one may not be firing. A plug wire may be missing its spark under load. If you let it sit and idle start by pulling one plug wire at a time to see if it idles worse, the one that doesn't make much difference is the weak cylinder. If your lucky that could be the problem.
An 88 has readable fault codes from the computer. If you turn the key on, off, on, off, on again all within 5 seconds you'll see the "check engine" light on your dash flashing. Count those flashes and post what code you see.
It will end with 5 flashes and 5 flashes(55). That will help also. After that let me know. You may need to do a compression test also if you get no codes. The distributor cap, rotor, and distributor pick-up all have problems depending on their age.
Checking the plugs(Champion RN12YC's) show if it's running rich, lean, or just needs replacing. Hope this helps you with your great car. Ok

OK Jonah, good luck and be careful, its a great car.

Take out the plugs they are probably all black from too much fuel, it sounds like its running too rich, and the black smoke proves its fouling plugs. After the fuel ruins the spark plugs they foul out and cause a rough idle and a shaking type of feel.
Before you remove the spark plugs do this. Turn the key to on, then off, on, off, then on and leave it on, watch the 'check engine' light. It will start to flash on and off, count the flashes. The diagnostic mode ends with 5 flashes and another 5 flashes. Code 55. Ok? I am looking for a code 11, or 13, or 22, or 51, or 52.
Let me know what codes you get, this diagnostic mode must be done within 5 seconds. As many times as you want.
I'm thinking the o2 sensor or coolant temp sensor may be bad. These would cause an overly rich condition so before we spend any money on plugs lets make sure of why its running so rich and with a high mileage car(and a good car also) many things could be wriong. Ok?
This is a 2.2L engine? 4 cylinder? We will try this now ok?