Buy some "real" degreaser or "purple power" from a auto parts store like Napa, make sure that it is not deluded already and that it is made for heavy duty cleaning (you don't want something that works on the kitchen as well) Also pick up a spray bottle, and some good tire wet and grab some change if your going to clean the tires at the same time your out.
Pour the degreaser or purple power into the spray bottle and take your car to one of those do-it-yourself car washes places with a high pressure nozzle. Begin by spraying the loose dirt and grease off your tires, then spray the tires with the degreaser and walk around the car and do all 4 before you go back to the one that you started with to rinse. Some of these places have a setting for tires and rims but just keep it on rinse, as the chemicals they have are not that great.
After you have removed the main grease and dirt a second application can be used if there is still a greasy look to the tire. A clean tire will not turn brown when you spray the degreaser on it, it should look completely dull when it is dry. Once it is dry and you can use a old terry towel if you want, then apply the tire wet as desired!Are your tires just nasty looking and have a thick coat of old tire shine on them? With this guide I will teach you how to get that nastiness off of them.