I grab the end of the pinion and try to move it sideways. I've collapsed the collapsible bushing before. The Pinion gets pretty sloppy.
Pop the cover off and catch what comes out in a fairly clean pan and take a look. Hopefully it doesn't look like shinny gravel.
I rag out the left over oil, then spray the whole works out with solvent.
Check all the gears you can see for dark spots blue/black or straw colored patches. Take a look at the spider gears for rounded edges or other obvious signs of excessive wear.
Put a pry bar under the ring gear and lift, if it moves much it's likely the carrier bearings are worn.
You can pull the rear drums and check for the outer seals seeping, a pretty good sign the wheel bearings are worn.
Disconnect the driveshaft, check the U joint. Then turn the pinion over by hand, you can often feel if something is wrong, like a rumble. If you remove the center pin, you can spin the axles pretty easy and feel if something is noisy. Then slide the axles out of the side gears. You can spin the ring gear and pinion and hear/feel if a bearing has a rumble, it turns pretty easy.
If everything is tight and doesn't show burn marks, I put it back together and add a pint of molybdenum to quiet it down some and run it.
The spider gears can be a little tricky to get back in again, remove them slowly and watch how they come out, check the thrust washers for excessive wear. If there is a locking bolt on the center pin, be real careful of that sucker, I've seen a lot of people snap head off.
