Grinding rear end

Here is the problem.When I make sharp turns at low speed or back up and turn,I hear this god awful squeel noice and grinding noise from the back end.Sounds like fingers in a fan blade kind of.I have heard many a rumor of rear ends in Z-28 not lasting to long.Here is a little history on the car.I have beefed up engine quite a bit,have aftermarket wheel and larger tire(made noise before,just not as much).Has 80k on origonal motor and tranny.I also drive like it is the last day I will ever have it,which at my present pace not very long.If it is the differential what is a good one to replace it with?Same for the axels?

Also recently had transmission oil changed and a tranny cooler installed,now under W.O.T. tranny holds first gear and slips going into second.All other shifts are fine.Fluid is full not overfilled or anything.

I had a problem that is almost what you described. I pulled the 10 bolt cap and found alot of metal around the magnet. Changed the rear end oil, went out an ran at the strip. I tore the Zexel Torsen up on the 2nd pass. The gear in the Z/T that is splined with the axle shaft was worn wear it meshes with the Z/T helical gears. Bottom line..
Pull the diff cap, look for metal. Spin each wheel while the cars up in the air. Look close at the gears on each side of the paddle.The (paddle) is what goes between the gears on each side and has a bolt holding it in. If you need a new Z/T SLP sells 3 series for $99.00

How difficult to do myself.Have good tools.

Not bad. The only special tool will be a torque wrench.
You might get a dial indicator to check end play