What is causing my kitchen drain to back up into my dishwasher?
It sounds like your dishwasher drain line is not installed correctly. You should have a loop so the drain line goes above the bottom of the sink before it goes to the dishwasher. Generally, the loop is attached to the bottom of the countertop itself.



Without the loop, when water is draining from the sink, it will naturally flow down the dish washer line. If there is a restriction in the sink drain, it will make things even worse, as the dishwasher drain is likely to the path of least resistance. 

The loop prevents water from draining down the line so long as the standing water level is lower than the high point of the loop. If your sink is almost completely full of water without 
the drain stopper in (to the point it drains into the dishwasher), you have an upstream clog that must be dealt with either way (you'll have the same issue with the dishwasher draining into the sink and/or not draining at all).