You won't get the black trainer card.
You will get another star added to your trainer card and it will change colour. You get the black one when you have all 5 stars. You get the others for:
-defeating the elite 4
-completing the national pokedex
-getting an 100 win streak at the battle tower
-winning a master rank competition in the Pokemon contest thing
-getting a platinum flag in the underground
When you get an 100 win streak you also get a gold trophy for your underground base.