When I run my dishwasher the dirty water backs up in my kitchen sink.

That has happened to me a few times. Running the garbage disposal always seems to help.

This calls for a plumber with an endoscope type of camera. He (or she) will use the camera to look down the drain pipe and find the clog. There is a clog and in my experience drain cleaners only work in television commercials.

Always make sure the disposal is empty before you run the dishwasher. Try using baking soda down both drains, about a quarter cup each or more. Follow with about a half cup of vinegar each down both drains. Let it sit and fizz for at least ten minutes. Follow with a gallon of boiling water between the two drains to rinse everything down the drain. If gunk is slowing down the drains, this should help open it up. I hope this helps. Good luck!

run garbage disposer if got one or see if the hole is opening going to garbage disposer from dish washer the same thing hapen to me theres a metal piece in garbage deposer that has to come out to let water out to other side if not that clean out bottom of dish washer

Did you just have dishwasher installed? I ask because this happened to my aunt and it was the connection and the length of the waste line. It was to long so it had a dip that caused waste or water to back up into sink. If this just started happening have you moved anything under the sink to accidentally push or move the connection? When you plunge it if it drains slowly does the same thing happen when you wash dishes by hand in the sink because then I would say it's a clogged sink that has caused the main drain line to be so slow that the water needs a place to go and backs up into your sink.

That was our exact problem. All of our drains were slow, including hand washing dishes, it would take forever to drain. Several times water backed up in the sink from the dishwasher. Finally, nothing drained at all. Called a plumber and they wanted to replace a 6 foot section only going to the plumbing stack And quoted $1,200. or to put an environmental grease eater, which they did, but where they poured it, it only sat in the elbow under the sink and that was not where the problem was. I wanted the drains snaked, but I guess they didn’t want to work that hard. I rented a 50 foot drain snake, and OMG, the drain lines everywhere were blocked completely. The stink was horrendou. My hubby and son worked on it for a week (both working), had to rip finished ceiling open, and replaced all the drain lines. Finally, for the first time in 9 years the drains work instantly. It was hard disgusting work, the whole house reeked, but am so very grateful hubby and son could do it. We saved a ton of money. This should have been a job for a plumbe but they didn’t want to do any work.

You have some sort of blockage - maybe in your dishwasher pipe

Have you checked your dishwasher's drain and filter?