While I was able to clean the dust off around the motor with the cover off the only way to clean the fan blades was to remove it from the ceiling. There were two flat headed screws on the right side in the photo above that had to be removed. They weren't in the metal but rather in the drywall itself with the edge of the fan assembly resting on them. The edge of the metal frame has two hooks that fit into slots in the ceiling. There's also a simple two prong electrical plug that needs to be unplugged as well. (I didn't see any reason to shut off the breaker to the bathroom but it might not be a bad idea to do so just in case.)
This was the fan after removing it from the ceiling. I ended up using a bristle brush attached to my shop vacuum to clean out all the dust in the blades. Putting it back in was as easy as removal. It definitely helped with the humidity and I'll be adding this to list of items to clean every few months to keep it from getting this bad again.
A few days after I cleaned the master bath fan I checked both the hall bath and the downstairs half bath. Both of them were slightly different in the way the cover hooked into the ceiling and the way the fan was mounted in the ceiling. Neither was bad enough to require full removal to clean at least.