Ok....
the woodgrain surround pops off it has clips all over, i think 4 on each side and 1 in the middle on the top and bottom. Put electrical tape on the end of a flat head screwdriver (so you dont scratch it) to wedge in between the woodgrain and dash. Be careful when you pop it out to not bend it though. Then use a 10mm socket and remove the charger section first then the climate controls. Unplug what you want to remove and clean how ever you need to.
May I ask why you can't just clean it right there on the tahoe? It seems there would be no difference unless you are actually wanting to take the climate control apart to clean it deeper.