Can I use Dish detergent to kill insects on plants?

Detergent no, liquid soap yes.

Soap has been used as a natural insecticide for years the trick is not to use too much or you will kill the leaves as well, the way it works is that it disrupts the insects cell membranes, and kills pests by dehydration.

Buy a liquid soap and not a detergent, something like Dr. Bronner's Pure-Castile Soap can be found in health food stores. mix 1 to 2 tablespoons liquid soap to 1 quart of water