Steam cleaner would be a last resort.
Go to your local supermarket, get some all purpose cleaner (liquid, any brand will do even the cheapest of the cheap stuff) and as many microfibre cloths you can carry.
Dilute the APC, dampen (dampen, not wet!) one of the cloths with the solution, scrub a section of seat, use a clean, dry cloth to dry the seat. Repeat until done.
If you're finding a cloth isn't lifting much more dirt, you may have to resort to a nail brush to scrub.