Oleander has dramatic flowers and flourishes well in hot, dry environments that wreak havoc on other plants. Unfortunately, it is extremely toxic and tough to kill. The leaves are extremely waxy and resistant to glyphosate and other powerful weed killers. If you want to kill the oleander, use a combination of cutting and poisoning, and even then you may have to repeatedly cut back and poison the bush.
Cut the oleander down completely and place the branches in a yard waste container.
Drill or cut a hole in each of the stumps and fill it with an undiluted glyphosate weedkiller or any strong weed killer designed for brush. Alternately, spray oleander directly over the whole stump. If you prefer to avoid toxic weed killers, boil water and pour it directly onto the stumps, covering each thoroughly. Avoid splashing the water on yourself or on nearby plants.
Cover the oleander with several sheets of newspaper, a black tarp or some other barrier to prevent sunlight from reaching the oleander. This helps prevent new growth.
Remove the newspaper or tarp and inspect the oleander stump for new shoots twice a week or more. When you see one, cut it off and spray weedkiller or pour boiling water directly on the wound. Continue to keep an eye on your oleander stumps until you haven't seen new growth in several weeks. At that point, your oleander is probably dead. Check on it at least once a season to make sure it doesn't come back.