Why does the rear end of my truck vibrate/shake when I accelerate from a stop, especially on an incline?

Sounds like you need to check your u-joints,, Or possible transmission mount could be bad,,, If it's only under a load that's what id check first,, And if it has a 2 piece drive shaft , then lubricate the slip yoke, when they get dry they will cause the same problem , All you haft to do to lubricate it is use a grease gun, their should be a grease fitting on the side of it,, If you check theas things and there is still a vibration, jack up the rear end put it on jack stands, put it in drive and get some one to give it some gas, climbe under the back behind the tire not in front of it,, shurly you know the reason why, and look at the drive shaft and listen for noise in the rear diff,,,

Check the rear universal first. Jack up the rear of your truck with a couple of stands and with safety in mind start the engine and put it thru its phases and see if you experience the vibration. This of course is assuming that your rear tires are well balanced.

maybe you have a transmission mount broke.

Shockes are going out. Better have em checked out.