What should be done about water under bathroom floor tiles?
Kinda hard to follow exactly what is happening there, but here are a few thoughts.  

First of all, the source of the water must be addressed first. If in fact it is coming from a window, then the window must be fixed, sealed or otherwise stop the leak.  Am I correct assuming the leak is coming from outdoors?

Second, any water under a floor is a problem. Although the tile itself will not grow mold, trapped water below the tile will effect the subfloor. Even if there is concrete board below the tile, there is still a wood floor under there somewhere. If the grout has cracked or become loose and you are seeing black coming through the grout, chances are good it is a diluted mold. The mortar, thinset or grout that holds the tile is now a hardened cement and will not dissolve easily with water. I seriously doubt black liquid would be created by either the mortar or grout. More likely is that there is water between the tile backer and some wood product causing the discolored water.

Last, any standing water under a floor is a breeding ground for mold or vermin like ants or termites. If not dried out, any rotting problem is just going to get worse and eventually compromise the integrity of the entire floor.