Cut back the stems of the Nerium oleander bush with a sharp machete, chainsaw or pruning shears. Don’t worry about making the job pretty because you are not pruning the bush. Continue to tear apart the bush until nothing remains but the stump. Throw the yard waste into a sturdy black plastic bag. Tie the bag shut and dispose of it through your local waste management facility.
Slip on a pair of rubber gloves, safety goggles and a respirator before drenching the exposed stump with a mixture of plain water and an herbicide that contains triclopyr and 2, 4-D, or 2, 4-Dichlorophenoxyacetic acid. Mix the herbicide and water according to the package instructions and continue to pour it directly onto the stump until it is completely saturated.
Cover the stump with a piece of black plastic or tarp. Do not use any products that allow sunlight or ventilation to the stump. The idea is to combat the stump, and root system, with herbicide while depriving it of oxygen and sunlight. Allow the stump to remain covered for at least one week. Check on the stump, and if it still shows signs of life, including new growth, inundate it with a second application of the herbicide.
After the Nerium oleander's stems are cut back, remove the root ball to eliminate the shrub. Dig up the soil around the plant's root system until it is completely exposed. Wearing long rubber gloves, lift the root ball from the ground and place it into a sturdy black garbage bag. Examine the hole for any remaining shoots before refilling it or planting alternate vegetation. Do not mulch or attempt to burn the root ball, as this releases toxic fumes that are potentially dangerous to humans and pets.