Buy a recommended herbicide such as glyphosate for killing the hackberry tree. The chemical is sold under different trade names.
Cut the tree down and make sure the stump is as level as possible as this will prevent runoff of herbicide. A level surface will also allow an even coating of herbicide.
Apply the herbicide immediately to the freshly cut stump making sure that the cambium layer or the area immediately after the bark is thoroughly soaked. These tree tissues will carry the chemical into the tree roots. Treat the entire surface on young trees with a trunk diameter of 3 inches or less.
Apply herbicide to 2 to 3 inches of the cambial tissues next to the bark on larger trees. The internal heartwood of larger trees is dead and does not need unnecessary chemical application.