If you expose the water to sunlight, algae and bacteria will remove/break down most things really quickly, the sediment will settle, fats, oils, soap scum will float and you can syphon from the middle of the tank/bucket, if you add some worm wee, yeast, molasses, or other goodies to it you can have a great little concoction for the plants.
Got a mate that has a $10 solar pump he puts in the bath with the panel hanging out the bathroom window.
4mm Aquarium hose leads the water into a 200lt plastic drum sitting on a post in the middle of the garden.
The drum has a tap and tap timer in the middle of the drum that leads to his fruit and vegies or he just fills his watering can from there.(There is another tap on the bottom to drain any sand or sediment that occasionally builds up)
Really cheap set up, and it seems to work great for them.
I climbed under the house and have diverted all the drainage pipes from the shower, kitchen and laundry sinks, and gravity feed then store all of our greywater in several 1000lt IBC tanks.
I have a constant bloom of algae breaking down any scum and 3 stages of filtration storage, removing the sediment and aerating each tank before pumping further up the line to eventually be gravity fed. I seed the water to start/continue a bloom at each stage that way the water doesn't stagnate and it is amazing good looking. Light green tinge, no scum or smell. I use it on underground irrigation on trees and vines, (not on our annuals or vegetable crops) but our soap/chemical use is really minimal.
It is amazing how much water we use in a day, and how much of it is normally wasted.
P.S. I didn't mess with the toilet, and my modifications may not be strictly up to code.....
Without it this place would be a dustbowl.