You need a degreaser, preferably a citrus based degreaser like the zep product found in lowes for 10 bucks a gallon (30%) Concentrate, or even better some Aero or any other 100% concentrate citrus degreaser from a janitorial supply house (we pay about 25 a gallon for this). This is for contact cleaning because you can't get to the bottom of the carpet.
You need a wet/dry vacuum with some good pull to it.
You need some soapy water or other cleaner for removing the degreaser residue after the degreaser is used to break up the clp.
So degrease it real wet, vacuum that out, degrease again real wet and vacuum that second time, then clean useing the cleaner real wet and a rough terry cloth type towel (for all of this, a clean one for each step) and vacuum that, then clean again real wet and vacuum one last good long time to get it as dry as possible especially the pad under the carpet. Just let the hose set in one spot to draw the cleaner up as much as possible.
Good luck.