I'm not sure how much it costs, but I'm with everyone who says do it yourself! You'll learn a lot about bikes that way and have a fun evening-length project.

I shouldn't contribute to the solvent wars, but it's always too tempting. Whenever I can, I soak and brush parts in undiluted Simple Green, rinse in water, and allow to air dry. If the parts are polished aluminum (which Simple Green will corrode enough to haze) or I can't/won't disassemble the parts enough to allow for a good air drying, I'll use whatever petroleum-based organic solvent I have handy (I just finished a bottle of VM&P Naphtha). I use Phil Wood grease for bearings, ordinary lithium grease for other greased bits, Phil oil for things like brake pivots, and motor oil for Sturmey-Archer hub internals. I'm in flux about chain lubes. But it's worthwhile to remember: it's really fun to fight over which solvents and lubricants reign supreme, but it probably doesn't matter all that much in most cases.

I'd say the trickiest bit is cleaning the hub shell, since it's connected to the wheel. The way I do it is to wipe out the old grease as well as possible with a paper towel and then actually go to my bathtub and scrub it out with Simple Green and an old toothbrush. It's also a nice opportunity to get the rest of the wheel (tire, rim, spokes, outside of the hub shell) nice and clean too. Note, however, that you will probably need to clean the bathtub itself after this, depending on the hygienic standards of you and those you share the house with.

With respect to flammability, there may still be some chlorinated solvents on the market. They are high-strength and not flammable at all, though I really can't recommend them due to their environmental/toxicity hazards. Basically all of them cause cancer, and personally I'd rather use mineral spirits/charcoal lighter fluid/kerosene/whatever. They're really not all that bad given adequate ventilation and very basic safety precautions.