Mow the grass to get rid of hackberry seedlings. The mower blade removes their growing tip, and they die.
Allow young trees to leaf out in the spring. Spray the leaves with brush or tree killer containing a 2 to 10 percent solution of glyphosate or a 60 percent solution of triclopyr. Spray the leaves thoroughly but not so much that the herbicide runs off.
Cut the tree down as near as possible to the ground. Chop or saw a V-shaped notch on the side you want it to fall, and a narrow backcut on the opposite side. Move away along a cleared path when the tree starts to fall. Pour broadleaf weed killer into holes drilled in the stump. Paint leaves of resulting shoots with a weed killer.