There are some simple, economical ways to flush a toilet with greywater. Here are two we recommend.
Easiest way: The bucket flush method. Lift the lid of the toilet. Dump greywater directly into the bowl. This will create a siphon affect and the contents of the bowl will empty.
Or, install a simple system like Sink Positive, a toilet lid with a faucet: flush water flows through the sink lid so you can wash your hands in the water as it fills the toilet tank.
Most other systems for flushing with toilet water are manufactured greywater systems, or high end systems. These systems are costly, economically and environmentally, as they rely on pumps, and chemicals or electricity to function. We discuss manufactured systems to flush toilet more here.