To clean it first start with a cool engine. If your mower has a fuel valve switch it off. If you can disconnect the spark plug too. Next use a leaf blower to blast away any old dirt and major debris. Now gently lay the mower on its side fuel cap facing up. Try to get rid of any junk on the mowers underside as well.
After sharpening both sides of the blade place it on the kits cone-shaped balancer. If one side dips below the other continue sharpening it until the blade rests level. Return the sharpened blades to the mower and reattach. If you have a torque wrench its a good idea to use it here. Often you need to tighten the blade mounting bolt/bolts to a specific tightness level. Typically measured in pound-feet your instruction manual should list these specs. For example my personal machine the Honda HRR216VKA between 36 to 43 lb-ft of torque is required.
Some lawn mowers such as the Craftsman 37705 have a washout feature. It is essentially a garden hose fitting on top of the circular blade cover. This allows you to pump water into the blade assembly while running the engine. That action is designed to flush dirt and grass debris right out.