As always, start by making sure nothing is holding a charge... Observe proper HV precautions!
TCE is primarily a degreaser. Probably not necessary here.
If you have access to a keyboard vacuum or other similarly precise *low power* vacuum cleaner with a brush tip, I would lean towards that. Second choice would be *gentle* blowing. I would not resort to chemicals unless you want to take the whole thing apart which sounds like an epic pain in the ass. Before you vacuum/blow take the tubes out so you can really get the vacuum/blower everywhere. While the tubes are out clean them off, because tubes should look pretty and shiny, but not for any reason of performance. If you care about the markings on the tubes be VERY gentle when cleaning, and avoid solvents unless absolutely necessary - you can probably get the tubes clean enough with just a few drops of water and a clean napkin.
Dust is not a problem until it forms a blanket thick enough to be a thermal insulator, and you are nowhere near that point. The big exception being if it is metallic dust from metalworking but I'm assuming (there I go...) that this is conventional home/office dust so, yea, it's just ugly.