The water used to fill a toilet's tank doesn't need to be as clean as that you drink, which is the idea behind reusing the water from washing your hands to later flush. The most direct way to apply this concept is the toilet tank mounted sink. You can get retrofit kits and there are even some purpose built toilets that integrate a sink. Alternatively, it makes for a
fairly straightforward DIY project. Not only is this idea more frugal with water, it serves as a reminder to wash your hands, which can be especially useful for young children. It also allows the possibility, for new construction, of a truly compact half bathroom. Of course, there's no reason one couldn't have a separate sink that drains into the toilet's tank, and there's also no need for it to be hooked to the toilet's fill valve either (this approach would allow for warm water washing, albeit in a less efficient manner).
As it is a common misconception, I should address the fact that water from your toilet valve is perfectly clean: it comes from the same water supply as the rest of your house. As to the reasonable concern that the water will be cold, I would observe that for bathrooms that have any significant length of pipe to the hot water supply, it will take a fair amount of time for the room temperature water resting in the pipe to be expelled anyway, so if water efficiency is your aim it's difficult to avoid. There are ways to get around this, yielding instant warm water, but the energy required to keep it at that temperature, as well as the additional equipment needed, should be considered.
