I have some experience detailing automotive interiors, and I would think the same methodology would apply here.
Depending on whether it is cloth or leather, it may be worth it to buy a dedicated auto cloth cleaner or automotive leather cleaner. These products are specifically formulated for cleaning, and have definite advantages over more traditional cleaning methods like soap, including not leaving residue (which attracts more dirt), being colorfast/safe, dirt repellent, etc.
A good option for sealed leather, as your computer chair would be, is 10:1 solution of Woolite liquid laundry detergent to water. The key here is to lather the surface with a brush dipped in the solution - shaving brush or large paint brush works - this will agitate the dirt and remove more contaminants than if you just apply with a cloth. Wipe clean with a wet cloth. Make sure to remove all traces of leftover detergent, or it will just attract more dirt. Lexus actually recommends this as cleaner in the owner's manual of certain vehicles with leather.
Steam cleaning is another option.