I think the best way to remove mud from water is reverse osmosis. But if your're think about doing it by hand than I would boil the water so the mud collects on the bottom. Then I would put a plastic wrap on top of the container while a small cup is inside the bowl. And then put a rock on the plastic wrap in the center on top of cup.
Explanation: The water vapor rises because of vaporization, and then collects as condensation and drips into cup-collection as pure water
I got this from a lab i did before, so all credits don't go to me =>