There are probably "panel clips" holding the door panel on. Go the the auto parts store and get a interior panel remover tool. It looks like a screwdriver but the end is flat for prying and is bent at like a 45 degree angle for leverage. They are only a few bucks. You will need to take out the screws that are in the arm rest before anything will come out. First thing is to take the cover off around the locks and handle. It is seperate from the door panel. It will look like its going to break but you can see where the clips are if you pry a little. Be sure to pry right where they are. IT will pop off and after all are loose you can pull that piece. This will leave you with the door handle and panel. Next you can pry a little around the door panel to determine where the clips are. There is a clip about every 5 inches or so all the way around the door. You have to pry pretty good to get them loose, they are just plastic. Some will probably break, if it happens they sell new ones at the auto parts store. Make sure like before, you pry right where the clips are. You can usually pry the panel out enough to see exactly where the clip is, put your tool right behind it. Once you get them all pretty loose, the door panel will have to be lifted about 2 inches straight up and it will come off teh door. This will leave you with the handle itself. It is all once piece and comes with the handle and lock mechanism. You can see the linkage back there and will have to undo it to get it loose. I "think" they are little c clips that slide off, just be carefull not to drop them. The assembly itself is held on by pop rivets, and they will have to be drilled out. The new assembly will come with new rivets. Make sure and not drill it out bigger than the hole thats there. Just go a tad bit smaller on the drill bit size and the rivet will fall out. Obv you will need a rivet gun to install the new one. Those assemblies are pos's and yes they break all the time.
edit### i just saw you had a silverado, scratch the assembly coming with the lock mechanism, yours will have electric locks. IT will just be the handle and rod on yours, no lock to fool with.