I'm at a loss to explain how water is getting into the overflow area such that it runs out the inlet. Perhaps your basin has an unusual configuration that allows leakage from the faucet to collect in the overflow area. Not only is this an unusual configuration, but faucets should not be leaking anywhere as well.
Even lacking a reasonable explanation, one thing is clear. The overflow outlet into the main drain is obstructed. This is usually through small slots in the top of the tailpiece that aligns with the overflow area in an interstitial space between the inner and outer basin surfaces. Grease, hair, and soap scum often build up around the stopper assembly that occurs in the same area. Some of this will end up getting through these slots. Attempts to clear the stopper area of debris can push more material into the slots, compacting the material into an impenetrable blockage. The main drain will work fine, but the overflow will only collect so much before it fills and stops working.
The best way to clear overflow blockage is to disassemble the drain assembly. Short of that, the slots can often be mechanically cleared somewhat with a stiff hooked wire or similar tool. Once loosened, the debris can often be flushed out by running water into the overflow. Drain cleaner may also clear the blockage, but I would be concerned the heat generated could damage the basin. Bleach will kill off the black mold that grows on this stuff, but will not help actually clear the blockage.
Once the overflow outlets are functioning properly, nasty stuff will no longer come out of the overflow inlet, but this does not solve the issue of how water was getting in there in the first place. It will still be flowing in there, but now it will pass into the main drain unnoticed.
