You can do it at home if you you're keen. You can get a reco'd LSD centre for around 300. It allot to explain on here and I don't have any pics because my diffs are IRS.
A quick rundown is you have to take off the back cover, remove the c clamps holding the centre in, the c clamps off the axles, take em out, remove the centre using a crowbar if it's stuck (diff stretchers are for girls
) then bolt the crownwheel onto your new centre and put it in.
The technical bit is the backlash. You have to check it with a dial gauge. Then if it's too high you have to add a bigger shim to the side opposite the pinion. Or vice versa.
Anyway none of that probably makes sense so if you're keen to learn best buy a book on how to build diffs. The principles are the same for pretty much all live axles.
You won't have to set preload if you change centres because you don't take out the pinion.
An easier option is probably just putting in a new diff that's already LSD. But like Bowen said it will probably be flogged.