Modern flat-screen HDTV monitors tend to have a fairly consistent set of mounting options which make it much simpler to figure out how to affix them into a cabinet. Mine had a stand which unscrewed with four screws, and also had a set of four screwholes farther up which would be the usual mounting point. But I decided to use the lower mounting block so I could affix the whole thing at one point. In practice I'm not sure if this was the right choice. It worked fine, but maybe using the others would have been easier. So I'll document what I did, but you might want to consider your own options.
My goal was to have it pretty much flush to the front of the cabinet, since the built-in black frame would look quite nice in the cabinet. So I cut a piece of wood to the right size to replace the plastic stand. At first I planned to just use a block of two-by-four, but I soon realized it would have to be so slender I could use the same screws through it, since I had no other screws I could be sure would fit securely. I ended up having to take a slice of melamine and slice it even thinner.
Once I had that, I screwed it into place, and cut a monitor shelf that was about eight inches deep, deep enough to go halfway through the cabinet plus some, but not so deep it would block cables reaching from the top to the bottom half. I brought them over to the cabinet and had someone else hold the monitor in place, centered and flush against the cabinet. Then I held the shelf in place and used a sharpie to mark where the mounting block sat on it. I took the whole thing out and used a bracket to affix the mounting block to the shelf.
Then I repeated the same process, with someone else holding the monitor in place, to mark where the shelf would go and where a set of brackets supporting its underside would sit. I then affixed those brackets, and finally, set the shelf on top of it and screwed it in both from the sides and from the brackets underneath.
This held the monitor in place pretty securely, but with it on an angle and only supported in its lowest few inches, I thought it wouldn't be a bad idea to add a bit more support at the top. So I put a few extra brackets in just to hold the monitor's top in place.
I'd left a little bit of gap on either side just to give me some slop, so I cut a few thin strips of melamine from scraps and mounted them to fill in those gaps. I had to use the Dremel to gouge out one of those to make room for the buttons that stick out one side. Some touch-up paint and that'll look like it was all part of the plan.