To do the job properly, take off all the hinges and catches first and keep them safe. I usually strip out the entire cabinet for cleaning, or as much as is practical.
I've had great results with a soap bar such as Astonish Stain Remover, or even one of the old style green bars of Fairy soap. If you use that with a bowl of warm water and either a sponge scourer pad (or nail brush if the dirt is badly ingrained) it should do the trick. Wipe off the soapy residue with a clean damp cloth and then leave it to dry. It leaves the rexine clean and with a nice feel to it. To finish, a very small amount of spray furniture polish (spray on the cleaning cloth rather than on the cabinet) will buff up a nice sheen when the case is thoroughly dry.
Everyone has their favourite method but that works for me.
Good luck!
Steve