I'm thinking about some type of floor buffer or maybe a carpet shampooer of some type. Might be able to rent something...
A lot depends on the source of the grime/stains - you should test for the best approach.
TSP (trisodium phosphate) would be one chemical to try. In the past, it was often used to prep outside surfaces before painting. I'd recommend testing any soap/chemical in some small, inconspicuous area first - and never mix chemicals - particularly bleach with ammonia.
other chemicals to try;
A vinegar/salt paste may work. It's slightly abrasive and creates a weak Nitric Acid solution.
CLR (the rust and lime removal stuff)
Ammonia (though that works best to cut grease - often, oil or grease is a 'binder' of dirt)
Hydrogen Peroxide (similar to a bleach, works well on many organic stains)
Citric or phosphoric acid (similar to CLR above, works for mineral stains,
Gloves, protective eye-wear would be a good idea for most of the above work. Ventilation shouldn't be an issue outdoors.
Pressure washing, if it works ,might still be a good approach. If the grout is so deteriorated that much is blasted away - well, consider regrouting. The result might be a like new surface!