I have been giving all my chucks a deep clean up in the last couple of days.
I agree with above about only taking the jaws out, but once every few years I take them completely apart. The first time I do a chuck I remove all the burrs left from factory.
Before starting, grab a piece of paper and draw the chuck with a mark for orientation to make sure they go back where they come from. For reference, I use the machined recess where the makers sticker goes on the front to show which way. Not doing this will become a headache trying to figure out which one goes where, at least on Chinese chucks.
Once the jaws are out, the yoke is on either side of where the unthreaded part of the screw is. Put the chuck face up and a few 1 inch bit of timber around the outside for clearance, then gently use a punch on either side of the yoke to force the pins out. Once they are down so far the screws will come out and you can tap the yokes the rest of the way and you can clean everything up.
I use spray can degreaser with a tooth brush to wash mine out as it's cheap, then compressed air to blow it off dry.
If you have kept them in order, putting them back in is just a matter of lining them up and tapping them back in. I well oil the holes and yokes before putting them in so there is no chance of rust starting.
Dave