To properly clean the inside of a piano, you need to open up the lid and see just how dirty things are. If there appears to be dust on the delicate parts, you cannot just start wiping them down forcefully. This can bring the piano out of tune, which can result in costly repairs.
Start the cleaning process by vacuuming the dust around the hammers and strings within the inside of the piano. Then, use a soft cloth to gently brush dust off of the strings. Use the vacuum to get rid of the dust when away from the strings.