Cleaning is one thing - disinfecting is another.
Clean with soapy water - a couple drops of dish soap in hot water will take care of dirt and grease - the two biggies.
A fresh filter is the best way to handle THAT collector of flaked skin and subtances oozed.
And the disinfecting part is simple: after you have cleaned, rinsed (couple of times) stuck in the nice new filter and started your new tub, add a couple dry ounces of Dichlor (chlorine) to the water and check the chlorine level five or more hours later. Still have a readable chlorine level?
Jump on in, the water's fine.