How do I get the Golden Kart pieces?

Solution 1:

From here: to get the GoldenKart, you need to "win every Cup with a single star on both 150cc and Mirror Mode in Grand Prix."

So, it looks like you've got a lot of races ahead of you!

If you are also looking to get the gold tires, that site tells you you have to "Beat every Staff Ghost in Time Trial mode at 150cc."