How do I know if my drive train, rear end or transmission is going out on my 2003 jeep grand cherokee

This rig is suppose to have limited slip in the rear end, if your getting tire spin in the rear then something has happen inside the ring and pinion, also as this is a Quadradrive system so if it detects whee slippage it should transfer power to the wheels that aren't slipping, it sounds like yours is not doing that. If it is shifting properly then you can rule out a tranny problem. So I believe you troubles are going to be located in the differentials and the transfer case, most likely an overhaul of the differentials and possibly the transfercase to fix the Quardradrive. HTH

Do you have any idea as to the cost for this type of work? Thank you for your answer

You could probably get the differentials done for about 5 to 600 bucks, but the transfer case is another matter, I've seen as high as 2 Grand go into one of these, maybe you have a 4X4 shop close to you that can get you a deal on one.