|"there are no user-serviceable parts inside LCD televisions or monitors"|
That depends on the monitor/TV & your skill level. There are individual circuit boards that can sometimes be replaced or if you're handy with a soldering gun, you can sometimes replace bad capacitors.
As for splitting the case, each monitor/TV is different - some have screws, others snap together, some use a combination of the two. You don't take the back off, you take the front off. Lay the TV face down on a table, remove the stand & any screws that you see, flip it over, then carefully try prying off the bezel/trim. Once you get it started, it should come off easily. After you get it off, remove the display panel, then you'll have access to the circuit boards.