Add a teaspoon of Borax to the final rinse when washing white clothing. Make sure the Borax doesn't have any clumps in it so that it will dissolve quickly.
Can you use it with bleach?
Be careful with borax if you recycle your graywater to the garden, (common here in Australia where we have droughts and water restrictions). Boron is a mineral essential to plant growth but too much is poisonous to plants.
I don't think there is any problem using borax in a load with bleach. As far as greywater systems go, I wish that was more common here! I think most people in the US just have it all go down the drain. I wonder how one would put a greywater system in? Maybe I'll research it. People in the Southwest are going to have to start thinking about these things... too many people and not enought water.