What are the cheapest and easiest methods for converting grey water (of the usual sources: kitchen sink, shower, laundry, etc) to potable water that is safe to drink?
This is a very open ended question. Much of this depends on the scale of operation, and your climate.
You specifically said grey water, so I'm assuming this means laundry, dishwater and bathwater. The chief grey components are going to be food waste, body dirt, and soap. There will be a minor component of oil or grease.
A good system would have a float separation of grease. Run the effluent into a large tank. Both input and output should be several inches below the surface. The tank should be sized to hold several day's waste water. Inlet and outlets should be designed to not stir up the water much. Using 4" pipe with lots of 3/4" holes is a good way to do this. (2 feet of pipe with 6 holes every 3 inches.)
You can check if you need this by taking some samples of grey water in a bucket, and see if a film of grease forms on the surface. If no grease forms, then you don't need this step.
After the grease separation, run it through a straw column. Build this with a couple of plastic barrels. Every 6" in height run 3 chunks of small diameter ABS sewer line through the barrel, alterting direction each level. These fit loosely into holes in the barrel, as you need to remove them periodically to add straw. On one side drill 3/4 inch holes every 2.5 inches. Fill the column, inserting the pipes as needed, hole side down. This keeps the column from collapsing into a sodden mess, and allows air into the column. Your effluent is sprayed onto the top of the straw collumn at the rate of a couple gallons a minute. You will need a small storage tank -- enough for a day's use, and a float switch to operate the pump. You need to either use a pump that can handle solids, or you need to put a filter before the pump. You will have to replace the straw in the column a couple times a year. Add to the compost pile.
Alternative way: Pile of straw on 4 pallets. Pallets are on a tarp. Tarp drains into edge of pond.
Run it to long, narrow shallow pond. The pond should be between 12 and 18" deep. Plant the pond with cattails. They will remove most of the phosphates from the soap, and generally remove harmful bacteria.
A second pond that acts as storage until you need the water. Having duckweed in this pond will act as a secondary biofilter.
To use the water first you need to filter it. Do this in several stages. First stage is a trash filter. This is a coarse mesh filter that keeps chunks out of the plumbing. Availalbe at any farm store. This filter should be on a float, with the filter itself about a foot to two feet under the surface.
Secondary filter is a screen filter. These are availble in mesh sizes from about 40 to about 150. You may need several in series with progressively finer meshes. Get ones that you don't have to disassemble to clean, but that you can just back flush.
Tertiary filtering can be either a sand filter or a disk filter.
At this point you need to start taking water samples and send them off for checks for bacterial contamination. Most pubic health offices in rural locations can do this for you. Contact them for sampling technique. If the bacteria count is too high, you can either use bleach or UV or ozone as a purifying step.
However the water is clean enough at the end of the cattails to use as water for your lawn and garden. Filter enough for whatever pumping system you use.
Caveats: Pond based filters don't work when they are frozen. If you live in a climate with consistend sub freezing or near freezing conditions, then you need to size your ponds to hold a year's water supply. You will also need to add a pump to circulate the water during the good part of the year. It doesn't have to circulate very fast. Once a week will do.
Numbers: Family of 4. 1 load of dishes (20 gal) 4 baths (150 gal) 2 batches of laundry (50 gal). Misc other 30 gal. Total 200 gallons. That's about 25 cubic feet. A 1 year pond then will need 400 x 25 (rounding days per year up) = 10,000 cubic feet. If the pond is 2 feet deep that's 5000 square feet. If you make the pond 10 feet wide it's 500 feet long. You can make the pond serpentine, so it fits into an area 50 x 100 feet. Or you can make it wander around the edge of your property as a water feature.
To circulate the water on a once a week basis during the thawed season you need to move 10,000 cubic feet / 600,000 seconds -- about a half a quart per second, or about 6 gallons a minute. You can use larger flows for the sunny part of the day and run it off a PV cell. There is very little height difference between one end and the other. Main power loss will be friction in the piping. You want low head pumps. Heating system circulation pumps, or aquarium pumps are candidates. Take the time to calcuate the power usage. It's going to run a LOT of hours.
If you have a moderate climate, you can put the water based bio filter in a green house. This will keep it active all year, or at most a much shorter down time. This means you don't need to store a year's worth of pond, but only a week or two. Make the green house twice as large as necessary, and attach it to the house as a sun room. This will provide some heat for the house, and give a pleasant place on a winter's day. Plastic sheathed hoop houses are inexpensive to build.
Further info: Wastewater treatment biofilter layouts compared. https://www.ncbi.nlm.nih.gov/pubmed/22508136 This one is for blackwater use on a municipal scale.
Wiki article about biofiltration in general https://en.wikipedia.org/wiki/Biofilter
Using wood chips and straw as a support for biofilms. http://www.sciencedirect.com/science/article/pii/S0144860907000581
This page has a forum for discussion aquaponics. For many a biofilter is part of the system to denitrofy the water from the fish crap. Not a direct comparison, but similar hardware.