If it's hard and cracked, then Rob is correct... you will need to channel Dr. Emmett "Doc" Brown.
If it is moderately flexible, then you may have some luck with Shift-It. Shift-It is mildly caustic, and has detergent and surfactant (ethyl glycol monobutyl ether) in it. Get yourself some Shift-It from Autobarn (
https://www.autobarn.com.au/kenco-shift-it-5-litre) - they have smaller bottles than the one in the link. It moves grease and grime, and is not too harsh on either you or the trim.
Pull the trim out of the car and give it a quick wet down (just enough to make it damp, don't drown it). Splash a little bit of the Shift-It onto the damp trim and scrub it in with a nailbrush. Work in the shade, doing a section at a time so that it does not dry. Rinse the Shift-It off with a clean bucket of water/rag... you would be surprised at the manky colour of the water. Repeat the scrubbing process with more Shift-It until the rinse water is relatively clear.
For grease/paint stains that did not come off, a little bit of prepsol can work wonders. At a pinch, thinners will too, though it can also discolour the vinyl. Go easy, test on the back of the trim first.
Once the trim is clean, you have three choices:
a) live with the resultant colour. Put some Armourall (or leather trim preservative) onto it to protect it.
b) Recolour the trim. Errol is right - the VHT trim paints have a good reputation. The centre panels on my wagon trim were resprayed, and came up a treat.
c) Selectively replace some of the trim panels.
Cheers,
Harv