How do I kill and remove an oleander bush without paying a landscaper.

Why not advertise it at the front of your property as Free to Collector - Bring your Tools? or words to that effect..........

Kerosene is highly flammable and toxic, not the best idea. Just dig it out. Vinegar will kill it but you will need to repeatedly drown it - and will take some time.

I knew a neighbor who was trying to kill a large Pampas Grass..he dumped a gallon of bleach in it and it was history.

https://www.gardeningknowhow.com/ornamental/shrubs/oleander/removing-oleander-bushes.htm

unfortunately, the majority of the roots need to be dug out. Chemicals may not reach all the way to the roots that can start growing again.

Depending on how large the oleander was in the first place, you can estimate the amount of roots you're dealing with. If you cut the stalks to the ground, dig out the dirt to at least 8 inches and use a pickaxe or loppers to cut the large roots (anything over an inch thick) , vinegar or any stump remover should do the rest of the work for you.

Dig it up. If you get the main roots it may not come back at all. If you pour kerosene on it, you may not be able to plant anything in that area for quite a while.

Keep trimming those small branches coming out so the plant won’t get bigger. Try to dig at least 3’ down and out to get all the bigger roots. If the stump offends you, put a bucket or pot over it for now, with another container sitting on top with a plant in it. And get it dug out when you can. ☺️