Okay, as stupid as this is, whenever I got used Commodore/Atari monitors or TV's from garage sales when I was collecting that stuff, if it was working, I wasn't too "fearful" of doing a complete cleaning inside.. I never wanted the dust/smoke/pet particles in my house whenever I used the set, so I always cleaned them.
I basically just took windex or 409 glass+surface cleaner (purple stuff) and soaked rags or paper towells and washed all around the tube, but never "touching" the rubber cap around the anode wire. I also got the first round off with the brush nozzle and a vacuum cleaner.
This got rid of 90% of the gunk.
Note: MOST TV's/Monitors, *IF* they are working correctly, DO disappate ALL of their built up charge backwards through the flyback transformer when turned off. if you hear a "cracking" sound when you turn the TV / Monitor off, then it's doing it's job.
You run the most risk with a) really old sets without this circuitry, b) broken sets where a fuse or other component is blown, that prevents the auto-discharge from happening.
Therefore when you're actually "handling" it, it's ALWAYS to be better safe than sorry, but my experience has been there is nothing to discharge because everything was working properly. BUT DONT TAKE CHANCES.
Also, I never remove the anode cap unless I have to--e.g. to do a cap kit or swap a tube. Otherwise, simple cleaning, leave it on and sealed, and you're (relatively) safe.
Note: Be careful being "too" vigourous in cleaning the tube (besides the normal warnings about breaking the neck) . Some people think it should be "clear glass" all around the tube just like the neck.... That is wrong. You see there is a special electrically conductive paint applied to the outside (as well as inside) of the tube glass. It *****NEEDS***** to be there to work!!! The tube is basically a huge capacitor, and the 2 layers of conduction, one outside, one inside, is how the tube works with the glass between them. Scrape off the outside paint, and you've just ruined the tube. So get the dust off it, but don't try to remove the paint itself, even if your rags "keep getting black" after you continue to rub it.
But it is always okay to remove the horrible "black dust" that the anode wire and outside of the tube attract. It's because it's a charged item that it attracts that type of dust more than anything else in a set.