Grey water filtration systems are one of those things, like pulling weeds or cutting firewood, that we might not necessarily want to spend time on, but may be best for us in the long run. Grey water systems can serve a few different purposes. The first of which would be to absorb, break down, and spread out chemicals, excess nutrients, and/or minerals which, in excess, may cause toxicity in our systems.
The main idea behind a grey water system is to take the output water from a sink, shower, washing machine, or some other system, which does not contain human or animal waste, separate large particles, which may clog the system, and run the water through layers of rock aggregate and plant roots, and send the filtered water on its way. Different mediums have different filtration properties, and depending on the needs and complexity of your system you may want to look into which materials will best meet the needs of your system, but for most basic systems, where only natural soaps and organic contaminants will be present, sand and any broken rock should do the job. Anyway, plants are the real heavy lifters in this system, and the aggregate serves mostly to slow the water and fill the roll of a rooting medium.
Different plants also have different filtration properties. Some are better or worse at filtering salt, and some are good for filtration of heavy metals or other things, but once again, for your basic system, any water loving plant will suffice. I recommend grabbing any water plants that you see growing in the local ditches and retention ponds.
Beyond that, this video should be all you will need to get your grey water system in operation.