Onan Gen Set 5000 very rough start

The black smoke on start up indicates too much fuel, so bad fuel pump, bad fuel filter or gummed up carb is not the problem. You indicate that it runs OK once it gets past the start up issue, so it could be a sticky choke shaft or a bad choke pull off diaphragm. If you can locate the choke pull off diaphragm, you can remove the hose attached to it and attach a longer hose and suck on it to see if it holds a vacuum. The choke pull off diaphragm will have a bent wire hooked to it that goes to a slotted lever. That slotted lever attaches to the choke shaft. Check to see if the choke shaft is free to move in its bore. Let us know if you find the answer.

Richard