Seems like you got a lot of sediment in your system. In this case a full flush is recommended. There are plenty of chemical solutions online but EPA regulations and such. Unless you are able to do proper pressure flush I would highly recommend getting a shop to do it as it is relatively cheap to do so considering the time and waste expenses and hassle.

The way I do my flushes is by draining out old coolant either by removing the bottom radiator hose or opening the radiator drain if there is one available.  Save old coolant so you can dispose of it properly.

Fill radiator all the way up with steam distilled water if possible. 

Drive till it gets hot then let it cool. 

Drain radiator again. (This helps get all the old coolant out)

Then fill it up with the correct coolant.   

That is how I do mine about every 100000 miles.

There should not be anything black in there. Coolant does not turn black.

You have a problem you need to deal with, and cleaning the system will not get rid of it. The black will probably come back.

Maybe you have rotting hoses.