If you want to roll your own, you need an LED and a Photodiode both facing in the same direction, into the water. Suspended particles in the water will scatter the light, reflecting some of it back onto the photodiode. The tricky part isn't so much the electronics, it's controlling the optical properties of the measurement environment. Once you understand the basic principle involved, you can certainly experiment.
On the other hand, If you don't want to re-invent the wheel, most modern dishwashers use them. Take your pick...http://www.appliancezone.com/SearchProd ... nSearch=GO
I especially like this one, and wouldn't mind playing with one some day:http://www.appliancezone.com/ShowProduct.aspx?ID=48034
Note the small size- the header connector appears to be a 4-pin, with 0.1" spacing.
You can probably buy one much cheaper than you can make one- even just to dissect and fool around with. It would still be a non-trivial challenge to get one of these to work, considering that you are unlikely to get much documentation with it.
By the way, such a device is called a Turbidimeter ("-meter" = to measure). I've used them before while doing control system jobs in the wastewater treatment industry.
The suffix "-stat" means to regulate or control.