The supply side is pretty easy as you can pick up some "splitters" that will allow 2 connections to each tap.
Where your challenges will come in, is with the drain line. A washer has a "pump out" feature so the drain is often located much higher than for that of a sink. Here you will need to tap into that line that will allow for a free flow draining option just a bit bellow the bottom of the sink...this should also include a "p-trap" to keep the sewer gas in check.
I agree with KMS. Your sink may fill up with water as your washer drains but it will be no different than a sink and Dish Washer set up. The Clothes Washer pumps out more volume than a dish washer. This is where you will find a problem.
In some older homes the washer drain line is a hose hocked over the sink. Often the end of the hose has an old nylon stocking to catch lint.