Coffee grinders are notoriously difficult to clean because most parts cannot be washed with water. And washing with a damp cloth often leaves coffee streaks everywhere and leaves your grinder looking far dirtier than it was when you started.
Because the coffee dust is dry, you can simply take a pastry brush or a clean unused makeup brush and brush the inside of the coffee grinder. It will easily brush all the dust and coffee particles from the inside and especially around the hard-to-reach grinder blades, which could be dangerous to clean otherwise.
You can also use a vacuum cleaner to get the dust using the brush attachment or one of the small mini vacuum designed to clean a keyboard or other small computer parts.