Always clean a moniotor with the power to it off/disconnected. The picture tube will hold a high voltage charge for quite a while under that rubber suction cup.
If you can take the monitor outside (on a nice day of course) you need only two items to clean it. A common cheap one inch wide (bristled not foam) paint brush and either an air compressor with blow gun attachment or one of those cheap cans of air from the office products store (used for blowing off computer keyboards, eetc.). You want to do this work outside as you'll be dislodging a lot of dust and some people are allergic to it. Be carefull with the paint brush around the convergence rings/tabs on the neck of the picture tube as you don't want to disturb their position (clean gently with the paint brush there).
If you don't want to take the monitor outside (or can't) then just use the paint brush and a vacuum cleaner (preferably a vacuum with a 'crevice tool' attachment).
As to the whining, visit Bob Roberts' site
www.dameon.net/BBBB/ and click on singing/whining monitor.