Those people who want to paint their monitor should probably consider using a paint that can actually handle being applied to a monitor. None of the three writeups currently existing inside this node actually suggest the use of the right kind of paint, which as anyone who knows anything about paint can tell you, is the most critical factor. You don't paint a house with nail paint, and you don't paint a monitor with house paint.
The easiest way to go is to purchase a specialty paint. The most readily-available one worth buying is the Krylon Fusion paint, which is a self-etching paint for plastic. This of course is an aerosol can. Pretty much any acrylic paint can be used, but it is then necessary to use a primer between the substrate and the coating, and also to rough the surface up first. Further, it's best to use the primer that's most similar to your surface color, so that if you do scratch your paint job, the resulting scratch will be minimally visible.
"Rough" is, however, a relative term. In this case, it means 500 grit sanding scratches. 500 grit is very fine paper, and will leave no visible scratches. It's mostly noticable because when you sand with it, glossy surfaces become dull. If you are not using a self-etching paint, you must rough the surface and apply a primer to ensure good adhesion.
Now some of you are probably shouting "bullshit!" right now, because you have painted something plastic, and the paint is sticking. That's nice, but move your monitor around a few times, and tell me if all the paint still adheres. If you want the paint to last then it is necessary to use the proper materials.
Regardless, when you're spraying paint, it's necessary to remove the monitor from the case. This is not a job for the faint-hearted, or those who like to stick their fingers into everything they see; the former will be too afraid to open their monitor, while the latter is not likely to survive the experience. Before I go any further, it is necessary to tell you that I will not be held responsible for any damage or injury resulting from following any of the procedures outlined here. There are probably better writeups on opening monitors than what I will provide here, but this should be sufficient information for those among us who are inclined to do this sort of thing.
Regardless of what else you do when you're messing with a monitor, the best piece of advice I can give you is to put one hand in your back pocket. It's very hard (though not impossible) to be killed by the electricity in your monitor with one hand in your back pocket. The killing dose of electricity is typically present even when the monitor is turned off and unplugged, and which can be retained literally for months afterward. Putting your hand in your front pocket is less useful because we do that all the time. We rarely put our hands in our back pockets (except to get something out) so it will serve as a reminder that what you are doing could kill you.
Also, spray paint is bad for you. It is something of a truism that breathing plastic dissolved in solvent is unhealthy. You can purchase a NIOSH-approved respirator meant for metals and organic vapors (which uses activated carbon to trap ions) for your protection, they're about twenty bucks for a disposable unit. I like the disposables better than the non-disposables in most cases, and they cost the same as just buying filter canisters for better brands, but you get a whole new face mask and straps. The respirator will last for about six months from the time the package is opened, whether you use it or not. The alternative is to stand upwind and hold your breath while you're standing in a cloud of paint. It's up to you, but I won't be responsible for lung damage either.
With that said: The majority of monitors are held together by just a few screws. Even very large monitors, like an old Sun/Sony GDM-series display I once painted black (it went to my Sun 4/260) often have a rear case held on by four (or less) screws, and the bezel comes off with perhaps another four. The best way to open a monitor is usually to set it on its face on a surface that will not damage the display (a towel on a flat surface is good) and start removing the shiniest screws that pass through the outer case. They are usually at corners or in the middle of long straight lines and/or large flat surfaces. Once you get the outer case off - it will come off the back of the monitor - you can find the four biggest, shiniest screws that go through the metal on the front of the monitor's tube and pass into the plastic of the front bezel.
Having removed both case and bezel, your next task is to wash them with soap and water. If you don't have a car soap, which would be my first choice, you can use about a tablespoon of ordinary unscented laundry detergent in a gallon or two of water and wash it with a sponge.
Now, if you are using a self-etching plastic paint, then just spray the monitor. I'll give spraying tips down a bit lower. If you are using a non-self-etching paint, then you will need to prepare the substrate to receive paint. You might be able to get away without a primer, but roughing up the surface is an absolute requirement. I separate each sheet of sandpaper into four quarter-sheets, which I find to be a more or less ideal size for sanding. One full sheet will probably be more than enough to sand your entire monitor, more or less regardless of what size it is. You don't want to sand it down, you just want the entire area to be painted to look dull. This will mean that it is covered with innumerable minute scratches which the paint will flow into, providing grip. You should be using wet/dry sandpaper, and you should be using it wet, so that it does not clog. Frequently dip the paper in a bucket of water, or continually allow water to run over the parts you are sanding. You can also soak the paper before use to ensure flexibility, this will help you properly sand areas like the inside corners of the front bezel.
Spray the case with primer, and then spray it with paint once it has sufficiently dried, following the instructions on the cans. If the primer and paint are not part of the same paint system, which is usually pretty obvious from the can design, then allow the primer to dry (and if necessary, cure) completely before spraying the paint, or the paint will very likely interact, which could cause color changes, crazing, warping, lifting, peeling, wrinkling, or a host of other problems.
Finally, some notes on painting. The best advice I can give you is to follow the directions on the can. If it says to make full passes at a distance of six to eight inches, do so. They know more about their paint than you do. The one thing you might consider doing is buying a fan-out spray tip, which can typically be purchased from the same place you get the paint. If you don't do this, it's almost impossible to get an even coat, which matters more or less depending on the kind of paint you use.
If you do get a fan tip (or buy paint that comes with one) then your primary priority should be to get even coverage. It may take you two or three coats for the primer, and as many coats again for color. Don't try to get it done all at once, you'll just be more likely to run the paint, and to get an inconsistent texture besides.
Also on the subject of the actual painting process; while I do suggest you get primer that's similar to your color coat's color, your paint is not properly applied if the primer color has anything whatsoever to do with your final color. Paint should completely hide primer.
Finally, if you want a glossy finish, apply a clear coat over the top, again following all instructions.
By far, the best thing to do is to get a primer (if necessary) along with the color and clear coat, all using the same paint system. If you have this option, I highly suggest you use it. When painting in this fashion, you do not allow the paint to completely dry/cure before spraying the next coat; each coat will chemically "bite" into the coat beneath it, providing the best possible adhesion.
The monitor goes together the opposite way it came apart after everything has at least dried completely, if not cured. (Lacquers, by the way, do not cure and are always "wet" so if you use one, just make sure that you wait as long as they tell you to wait, and that the surface is dry.) You can spray another similar piece of plastic and use that as a test sample to figure out when the paint is dry enough to spray over, which is generally when you can't leave fingerprint marks in it.
If you are having an orange peel problem (short description: the paint has the same kind of texture as an orange peel, more or less) or just want an extremely glossy finish, you can put on a whole bunch of clearcoat, let it cure (or dry for lacquer) and then sand it with 500 grit, then 1000 grit, and finally 2000 grit; then you just take a terrycloth towel with some rubbing compound on it to the thing, and polish until it shines.
Aside from the specific equipment used, this is the same process used to paint cars, and should result in a paint job that will outlast the rest of the monitor in much the same way that the monitor's factory paint job typically will. It will look clean, and professional, and best of all, it requires no masking of your screen. The simple truth is that masking causes problems, whether it's a line at the edge of your paint or leakage under the tape, and if you can avoid it, you should.
And just keep in mind, if you zap yourself, I take no responsibility. Have fun!