Mow the grass with your lawn mower as you normally would. However, set the mower blade to its lowest setting for maximum grass removal.
Saturate the grass with a garden hose.
Place sheets of overlapping newspaper over the area where you want to kill the grass, and then saturate the newspaper with water. Place a second layer of newspaper over the first layer, and then saturate that layer. Do this until you have seven to 10 layers of newspaper.
Place mulch over the newspaper. The mulch will hold in the moisture, block out the sun, and kill the grass within two months.