Level42 is very carefull. That is good. 'Better safe than sorry' is the best way to be sure that you will not damage anything. However hygiene is also necessary, especially for potentiometer and mechanical objects, also to be able to access parts and actually be able to see what part it is, and not getting your hands dirty ofcourse.
A very dirty monitor chassi I personally wash just like any PCB or dirty plates. I use water and soap and a normal paintbrush. Sometimes it may be necessary to desolder certain parts, but most chassis just need a good rinsing under hot water. If any grease is present under the anode cap I remove it with normal white spirit, or gasoline. And always, before washing a chassi take a photo of any paper labels on HT unit, and other if present.
Drying the chassi is the critical moment. I use to stand the chassi ontop of my Electrolux vaccum cleaner for 10 minutes (warm air coming out there), then I blow 'compressed air' (with my mouth) under components I feel are tightly mounted to the chassi, and then I put it back ontop of the vaccum again for 10 minutes. Ofcourse real compressed air would be best to use now, but I have no access to it at home.
The CRT I clean like any other window, with window cleaner and soft cloth. The back of the CRT and neck with yoke, I just use a dry brush to dust off the most, holding my vaccum hose to eat the dust as it flyes. If necessary I then use a damp cloth or white spirit to remove any sticky stuff. The 'dag' (conductive paint) I clean only with a damp sponge, then dry it with a soft cloth. Never use abrasive sponges or scrub this 'dag' off. It must be left intact.