If it can be done with raw sewage and garage waste such as motor oil and radiator coolant, I assure you the grey water going down my sink will not be an issue.
https://www.youtube.com/watch?v=49otYP9nnkQ
As I noted in another thread, there is no reason to filter out Polio or Cryptosporidia when you don't have those things on your hands or body. Therefore they do not get washed down the drain via showering or hand washing.
Countries such as Singapore, Australia and Namibia, and states such as California, Virginia and New Mexico are already drinking recycled water, demonstrating that purified wastewater can be safe and clean, and help ease water shortages. Of course, these facilities operate on a large scale, but there is also much more to filter as well. The technology is sound and no need to re-invent the wheel. Just copy what the environmental groups, natural disaster suppliers and militaries around the world have been doing for decades.
I own a 0.1 Micron (Absolute) as well as a 0.02 Micron (Absolute) water filter that I use for back packing and hiking. Both are small enough to fit in your pocket. With a vehicle platform and a bit of research, a proper recycling system should not be difficult to contstruct using COTS technologies.
EDIT: pre-washing of your hands to remove grease from a car repair for example, is obviously a wise decision. I only use Dr. Bronners Castille soap, which is safe and non-toxic. A mate of mine uses it instead of toothpaste. Take a bite out of your Dial soap and let me know how it works out for you. Billions of dollars are spent each year making water safe to drink or discharge into local streams and rivers. There is no reason you can't take advatange of that tech.