China is a dictatorship that is spending large sums of money to become "dominant" in space, whatever that means. There is already competition for orbits, frequencies, etc. in Earth orbit. It would not surprise me if China and the U.S. have plans to land in the same places on the moon and use the same resources.
That could work out like Antarctica, where there is space to set up multiple facilities and no shortage of the things that Antarctica has.
But the Moon could be a different situation, if there is only sufficient water for a small colony.
However, I am not seeing how China could claim parts of the Moon, given that they are signatories to a treaty that says no nation should do that. And, how would the enforce such a claim. War on the Moon would precipitate war on Earth's surface.
So, at worst, we will probably get in each other's way if we don't have joint programs.
And joint programs look even less likely now that we have the current experience with Russia and the IST codependence. Do we really want to be commited to a colony program that depends on China when they decide to invade Taiwan?