Lay black plastic sheeting over the area of lawn that you are eliminating.
Place heavy rocks around the edges of the sheeting to keep it in place.
Leave the sheeting in place for around six months. The hotter the weather, the faster this will work.
Remove the sheeting and turn over the top 12 inches of soil with a hoe. Break up any clumps of dirt.
Work in compost or another fertilizer if you aren't going to plant something else in this location.
Lay 4 inches of grass clippings, topped with 4 inches of newspaper and topped with a final 4 inches of compost.
Water the layers weekly to keep the organic materials moist and encourage them to break down.
Leave the layers in place for four to six months.
Plant trees or shrubs by digging a hole through the layers and planting the root system. The organic layers have the added benefit of adding nutrients, deterring weeds and conserving water.