Get some sort of carpet cleaner/upholstery shampooer to lift all the dirt and bacteria out of the seat.
If you want to do the 'super cheap and maybe just test it a little before I buy/rent something,' do the following:
1. Get an iron and a damp washcloth or towel (or shirt. something you don't mind ruining).
2. Place damp washcloth on an affected area (like where there's a stain, for example)
3. Lay iron on wet washcloth and let sit for 30 seconds to 1 minute.
The heat from the iron will pull the moisture and dirt from the seat into the cloth/towel.
4. Set iron aside somewhere safe
5. Lift washcloth from surface and marvel at the black, icky crap that was lifted from the seat.
I used to have to do this on carpets in nursing homes. Old folks would cruise around with their cranberry juice, spill it, and here I come with a little carpet cleaning chemical*, damp scrap towel, and iron to pull the stain up/out.
Anyway, if this little test works then go rent a machine because it is far more efficient and you're less likely to burn yourself :p
*You don't necessarily need a chemical to start with. If you find this works, go out and rent a carpet cleaner with the nozzle attachment (Think Blissel Brand. You can rent for $30/day from Orchard or Ace, probably Home Depot). If you use a chemical with the ironing method, you'll apply, agitate with a brush, apply towel, iron, wait, remove and review. Heck, your 'chemical' could probably be baking soda, a little water, and scrubbing an area with a toothbrush. Again, super tedious. Go get the machine w/nifty sucking nozzle and go crazy.
Hope this helps!