black smoke in exhaust means it's not burning all the fuel being injected into the cylinders.
So yes it could be a timing issue.
Since the previous owner claims he had the injectors and the pump rebuilt you may want to double check the timing.
The Fuel system diagnostic chart list these possible causes for black smoke:
- Fuel injection timing
- Poor compression
- Fuel injection
- Fuel injection pump

So basically you'll need to verify the timing for the fuel pump is set correctly, then check compression and if these check ok then your looking at injector or pump problems and will have to remove them and have them checked and possibly repaired.
It is possible that whoever did the rebuild on the pump and injectors did not do it correctly.