As an overview conceptually, to set backlash on a differential with shims like the Sterling, you set the carrier bearing preload by adding or removing shims (circular washers) that wedge between the carrier bearings and the housing on each side of the carrier. Most people simply keep the existing shims and shim count when removing the existing carrier and reinstall them right back in and then simply swap them from the right to the left to transfer the carrier right or left (bringing the ring gear closer or farther from the pinion) to adjust the backlash.
To measure backlash, use a magnetic base with a dial indicator set up tangent to the ring gear on any tooth face. Holding the pinion PERFECTLY still, and applying very little pressure when "bumping" each side of the play in the gears you can read the amount of play in thousandths of an inch. Depending on whether or not you are using new bearings and gears (would set a bit tighter to compensate for new parts seating to one another) or setting up an existing set with a new carrier for instance (would set things a bit looser as everybody is pretty well already where they're going to be), you can dial in the proper amount of backlash by adjusting the shims from the right to the left to decrease backlash, and from left to right to increase it. Adding a shim to the pack will increase the preload.
Gear manufacturers usually provide specifications for most common axles, reflecting ring gear bolt torque, pinion bearing preload, carrier bearing preload, backlash, so on and so forth. Yukon being a good source.
