Precipitation of water-insoluble materials from shower or laundry greywater
Your ambitious project is not that trivial. Soaps are of several types. 
a) If you are talking about salts of fatty acids, the classical soaps, then one can precipitate them out with calcium salts. Many other cations form insoluble salts with soaps. Now if you are trying to purify laundry water, dishwasher water, shampoos, liquid soaps, nobody can remove those surfactants from water by simple means. They (manufacturers) have specifically designed them so that they never precipitate out in water, otherwise your clothes and dishes would always have a layer of scum on them. Potassium ions may form ppts with certain surfactants like sodium dodecylsulfate but forget about ppt'ing out these typical surfactants from water used in ordinary household. 

b) Alum can quickly help you in cleaning all the particulate matters in waste water, like magic. 

c) Filtration on beach sand beds can remove small amounts of certain surfactants from water. 

d) Filtration on activated carbon would be the best to remove surfactants.

Well it turned out to be more complicated than you desired. There is no single cure and certainly this water is not good for mopping floors because it will spread nothing but germs from using laundry water and shower waste. You can think of solar distillation as an alternative. That distilled water would very clean.

Note that in certain countries, household waste water including the sewage water after doing natural bacterial treatments is used for watering gardens and small farms. In one large pit, all wastewater is allowed to stand in open air, far away from residential areas. In the first ou all solids settle down, next pit further purifies via same settling and bacterial oxidation process, finally the supernatant water is transferred to third pit, which is now clean enough that gardens and small farms can be watered with it.  Of course, this water is rich in nitrogen and phosphates (from human and laundry waters) which is good for plants.