I had seen this trick somewhere on the internet, maybe Pinterest, awhile back (I wish I knew where so I could give credit). It works AMAZINGLY well.
My kitchen sink is the standard white porcelain kind and is many years old. It seems that the original finish has come off and it has pock marks and divots all over that collect grime. The thing stains really easily and always looks dirty. If your sink is like mine, here is what you can do:
First, clean the sink the best you can. Then spray it with some sort of household cleaner that has bleach in it. While the sink is still wet, line it with 1 or 2 layers of paper towels, then spray again. Make sure that the paper towels lay flat along the sink and that you get them thoroughly wet. Let sit at least an hour (preferably overnight). Remove the paper towels and voila! Now, it isn’t perfect but it is a whole heck of a lot better.