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.
Place a drip pan or other container on the side of the mower with the filler. Carefully tilt the mower so oil drains out and into the pan. Discard the spent oil into a proper disposal vessel. Now slowly pour fresh oil into the mower. For the specific type of oil you need check your instruction manual. Make sure to add only as much as your particular model requires. Remember to let the mower sit undisturbed for a few minutes so the oil settles properly inside the engine.