Before you go changing injectors, check your fuel pressure. You may have a bad regulator which will make you run rich, causing stalling and black smoke, BUT with the back fire from the exhaust it could be an ignition problem. You may want to check your ignition coil as a possible suspect. Usually when they go bad it will start and run until it heats up and shorts(could be seconds, minutes or hours). A bad ECT, TPS or MAF could also cause the same symptoms and it may not throw a CEL because the car wouldn't be running long enough to. Plugging it in to a diagnostic computer (NOT a code reader) should point out a dead or dying sensor in the short time the car will run.
