I use westley's bleach white for tires. To get the build up off. I have used a brush the has nylon brissels on the outside and steel ones in the middle. take your time and it will come off. Use a small circle motion. Then from now on, never spray your tires unless you have cleaned them first. During winter months I don't put goo on the tires, but maybe once every couple of months. this lets the old stuff wear off so there is no build up. Meguires gloss tire shine, if you put it on once it last through about three tire scrubbings before it starts to lose it's shine. But the worst thing you can do is not clean the tire before gooing it. Over my 13 years going to car shows I see these people who don't wash their car because it's just a show car that never gets dirty then they put spray right over the dirt trapping it there for all to see a glossy dirty tire.
Another thing to try if it won't come off with soap and a brush is to use like a Brillo pad or SOS pad lightly around the tires not the wheels because it will scratch them.Also you can pick up Eagle One A2Z wheel and tire cleaner for when you wash your car just spray on the wheels and tires then hose off the dirt.Megiuars also make a tire shine called High Endurance that should last a while to keep the tires looking good.
My tires shine for atleast a week straight, a little longer if the weathers nice and I don't hit dirt. When I wash my foci I use a hard bristle brush, then spray this cleaner made by Black Magik, then spray it off. When it's dry, spray lightly Rain-X wheel protectant over rims. Then take Black Magik Tire Gel and put it on an applicator and rub it on tire. When it's dry, it looks sweet. Best shine I've gotten yet. As far as Rain-X, it makes cleaning the brake dust off easy. Next brake job, I'm switching to Ceramic Pads. NO DUST!