We just moved into our home (build 1967). Our kitchen started clogging right after one use of dishwasher. My wife mentioned that when she used the dishwasher, the water came into the kitchen sink (is it from the dishwasher?). Since then the kitchen sink getting clogged and it takes forever for the water in the sink to clear up. I've tried pouring entire bottle of liquid plumber but of no use. I've no clue what to do. Is there something that I can do on my own to resolve this problem or do I need to call a plumber?

I'd highly appreciate your expert comments/suggestions.

Is there something that I can do on my own to resolve this problem or do I need to call a plumber?
You can remove the "J" bend from the trap and snake the drain line,(see image). You will run into a 90 degree bend about 8 inches in but once around that you only have to put out 6 more feet of cable. After, flush with hot water. Good luck, Tom

Dishwasher drains into your sink plumbing: either into garbage disposer or directly into the sink drain via branch T fitting. As Tom correctly points out in the previous post you have blocked drain. It shows when D/W runs because it discharges lot of water in small amount of time. Since you have restricted drain, water accumulates in the drain and backs up into your sink.

Here it is a good example why you should have "high loop" on D/W drain or Air Gap installed. If you don't have loop or Air Gap - than D/W may fill up with dirty water when sink is used under these conditions.