It all depends on your camera. If it's a digital SLR with interchangeable lenses it's not too difficult - if you trust your own ability. If it's a fixed lens camera it has to be sent away, but that leaves you with the problem of working out how the muck got in there in the first place.
If it is just dust in an SLR I find a very useful piece of kit is the Giotto blower. Put the camera in mirror-up function, hold it upside down and blow away the dust. An E-Bay link is:-
http://www.ebay.co.uk/sch/i.html?_nkw=giotto+blower&_sacat=15200&_odkw=&_osacat=15200&_trksid=p3286.c0.m270.l1313
Unfortunately it's not always simply dust. Sometimes the muck is moisture borne and blowing won't shift it. This is a particular problem working outdoors in less than favourable weather conditions. I have three SLR bodies and have been cleaning them for years now using the swab and methanol treatment as described in Tom's post - this being the method normally adopted by the trade. However, I must stress that care and a steady hand is important.
I don't know what the cleaning charge is these days from a repairer, but I'm guessing it must exceed £20. One company I can recommend is the Canterbury Camera Centre. Not only do they offer a good personal service, they know what they're talking about.