A while back, my kitchen sink drain started getting clogged so when I would run the dishwasher, the water would back up into the sink. It was going to take a while before it could be fixed, so I had to figure out another solution. I didn't want to wash all the dishes by hand in the bathroom sink, so I figured out a way to still be able to use the dishwasher.
Under the sink, there is a hose that runs from the dishwasher to the drainage pipes or disposal. This is where the water is drained from the dishwasher. I found if you unhook this hose from the disposal and put it into a bucket or a wastebasket, you can run the dishwasher and the water will drain into the bucket. Then you just have to empty the bucket in a another drain that is not clogged such as a bathroom tub.
It is not the most convenient because it usually uses at least 5 gallons of water per wash so you have to keep an eye on the bucket throughout the cycle to make sure it does not overflow. I found that having 2 small buckets works best. It usually does not drain more than 1-2 gallons at a time so as long as you stay on top of it, it should be able to handle the water.
Check the instructions for your dishwasher to see how many gallons it uses per wash. It can depend on which cycle you choose.
Also remember not to use the sink when you have unhooked the hose underneath as water may get out from the hole where the hose has been disconnected.
Overall I find it works well and allows you to run the dishwasher even if your kitchen drain is clogged.