Nissan Skyline

I have purchased this skyline and had it all dyno tuned but when the car is cold in the morning I have to pump and pump and pump the accelerator and then when the car does start a big puff of black smoke comes out the rear. It runs rough for about 30seconds and then comes good. When driving the car under high RPM it seems to pop and miss and if I turn the car off for more than half an hour when restarting I have to pump the accelerator 5-6 times to while turning the engine over to make it start. When I spoke to someone about this they told me it sounds like a fuel injector leak as if one was leaking it would de-pressurise my fuel rail therefore dropping fuel into the motor therefore when you start it its hard to start and blows black smoke?? Just wanting a second opinion please.