First, get the vehicle up in the air. Remove the wheels. Pictured here is the removal of the brake caliper. Remove the two 15mm bolts that hold the caliper bracket to the axle housing. Then pull off rotor and remove rotor.

Next, remove the driveshaft. there are four 11mm bolts. Leave the car in park to loosen and remove the bolts and strap. Then put in neutral and turn the driveshaft until you can get two the other two bolts. Put back in park. Loosen and remove the other two bolts and strap.

After removing the driveshaft, you'll see the pinion nut. It takes a 1 1/4" socket to remove. This won't be easy if you are doing this by hand. An impact gun will do this. Another way is to remove the driveshaft before the brakes. then use the e-brake, or have someone hold the brakes with their foot on the pedal.

Remove the cover. There are 10 13mm bolts for this. Two of the bolts are longer. These two are the ones with a metal hanger that holds the e-brake cables.

Rotate the carrier until you find the 8mm bolt that holds the axle spacer in. Remove it, then take out the spacer.

There are C-clips that hold the axles to the carrier. Push in the axles and then remove the C-clips. Arrow points to where they are. When you push in the axle, they might just fall out. Don't worry, they don't go far. Just reach in and take them out. Then slide the axles out of the housing. 

Here's what the C-clips look like and how they just slide over the end of the axle.

Remove the carrier main caps (see arrows). This takes a 5/8" socket. I got a little lazy and didn't take any pictures of the carrier out of the housing. You'll need a pry bar to get it out.  What I do is take a 3/4" wrench and attach it to one of the ring gear bolts. Wedge it against something, then use the pry bar to get the carrier out.

Pictures I didn't take was of the removal and install of the ring gear and pinion gear. Once the carrier is out, put the cover back on with one bolt. When you go to remove the pinion gear, this will keep it from flying out the back. What I do is use a big hammer and put something against the pinion gear where the yoke is. Then hammer it until it pops out.

Remove the ring gear by taking off the bolts. These are left hand thread. Takes a 3/4 socket to remove. To get the gear off the carrier, I leave two bolts opposite each other slight on the gear. Then tap the bolts. This will aid in pushing the gear off the carrier.

Install the new gear on the carrier. With left hand thread bolts, the recommended torque is 80 ft-lbs. Torque in a rotating method. Either clockwise or counter-clockwise doesn't matter. Move from one bolt to the opposite one from it. Then move to the one next to it. then jump across from it. Continue until all 10 bolts are snug. Then torque the bolts. This will ensure you don't bind the gear on the carrier and break it when trying to seat it to the carrier. 

This is a pic of removing the pinion bearing from the gear.

Now, install the pinion bearing on the new gear. Once you've pressed it on, pump the press some more to add pressure to it. then tap the press shaft. you'll notice the gauge start going back to zero psi. Repeat this until it no longer does this. This ensures the bearing is completely pressed as far as it will go.

Install the new pinion gear without the crush sleeve for now.  Tighten the nut just enough to put some resistance when trying to spin the yoke and pinion. Then install the carrier with the new ring gear. Check the backlash. You do this by holding the pinion still and moving the carrier. Should have about .006"-.010" total. This shows the dial indicator when checking for backlash.

  Once you have this, time to use marking dye and check the mesh of the pinion and ring gear. The pattern you here is the type you want to see.

When I had Motive Gear 4.10 in the rear, I had to grind on the spacer to clear the teeth on the ring gear.