Preferably wear gloves and face mask when handling the mattress initially. Vacuum it well, as well as the bed base. Light mist of pre spray (enzyme if poss as there are usually lots of organic nasties on them) and have some urine treatment and rust remover handy as these usually have one or two errant stains. Agitate well with small brush and leave to dwell for about 15-20 minutes. If using a standard hand tool, lower the pressure so as not to overwet. Move slowly over the mattress in long stokes in one direction and two dry passes, mist extraction fluid over the seams and then extract to avoid over-wetting. Use your usual neutraliser in the extractor with a little deodoriser to make it smell nice. Stand up to dry to get better airflow, don't dry on floor or carpet as you'll only make these wet and may get some transfer (I sometimes use an old sheet if there is a hard floor).
One other tip - check for bedbugs first!!