The turbocharger has two halves, if you like. One's silver, the other brown. The silver side is the compressed air intake side - inside is an impeller, that sucks air in and pumps it out the rubber hose up the side. The brown side is the exhaust - or driving side. Inside is a turbine, that is spun by the hot, fast-flowing exhaust gas.
The exhaust gasket is on that brown side. It will have a black DUST near it. It will look like dust.
If you see what appears to be black texta marks on top of the silver side, it's probably the hose that connects from the turbocharger to the baffle which is a small silver round thing attached to the side of the motor. It's right near the label "turbocharged air" in the picture in my garage.
If it's black oil dripping off the bottom, then you may have a seal leak. Oil (and coolant) flows through the turbocharger in order to lubricate and cool it. If either are visibly leaking, you have a problem and the turbocharger really needs to come out for a service. Be prepared to get some new studs for the turbocharger mount, I had a siezed one in mine.
As for your symptoms - they could also be a failing bearing in the turbo. It happens - usually because the engine is shut down a little too early after a hard run, the turbo is hot and because the engine stops, the oil stops flowing through the bearings, allowing the hot turbo to cook the oil. It forms a hard varnish on its surface and begins destroying the bearing surface. After that, it's fairly inevitable that the turbocharger will start to make odd noises, then wobble and screech, then go
BANG
leaving small chunks of turbocharger to fly rapidly towards the intercooler. And that device, if it does nothing else in its life, will protect your engine from swallowing those bits, so you lose a $1400 turbocharger but save $10,000+ in engine repairs.
I hope it's not bearings. The simplest thing to check is boost leak, followed by exhaust gasket, then it gets harder.
To inspect the turbocharger (cold engine only!), you have to remove the air hose from the front and, with VERY clean hands, try spinning the impeller. It has to turn smoothly, if a little reluctantly from the oil in the sleeve. It should spin without any gritty feel and come to a stop fairly quickly. The shaft should NOT wobble at all. It MUST be tight. If it wobbles, the impeller could make contact with the housing and at 100,000rpm that's going to get really ugly really fast.
Let's hope it's a boost leak.