Jack and support the rear of the car so that you can get underneath to the diff..... If the diff casing is anything like the non LSD casing then stick a shallow tray under the diff and remove the plug at the bottom of the diff and allow the lubricant to drain.... I'm guessing that like the non LSD the plug will have a little magnet to collect swarf, so clean that with a cloth.... Once drained, refit the plug and remove the filler plug which should be on the side of the diff about half way up..... clean that off and refill the diff with the correct grade lubricant (SAE 90 GL5 for lim slip diffs) The diff is full when the lube is up to the level of the filler and a small quantity of lube trickles out. Refit plug, wipe any surplus lube from casing, lower car to the ground and go and have a cup of tea...
Tips, you may find that the filler is tricky to get to, so buy your oil in a bottle with a spout and short length of filler tube.... I have an old one in the garage for such an occasion.
If the lube is cold and thick, stand it in some warm water for a little while to make it thinner and it will flow a little better.
Dispose of your old lube by pouring it into a bottle and taking it down to your local recycling area and don't pour it down the drain.
Shove some cardboard under the work area to ensure that the oil doesn't spill onto your drive and stain it
Use axle stands not bricks to support your car as instant death MAY occur
Use a dedicated oil pan for collecting the oil and not your wifes grill pan as instant death WILL occur