Two separate areas here- A. KILLING the stump, and B. REMOVING the stump.
Now, to an old explosives engineer, there is only ONE way to remove a stump- but you neighbors will gossip about it for a while.
However, you CAN kill off the stump and leave it- will take a few years for it to rot away on its own. As said- use a spade bit on a drill- drill a few holes down from the top as deep as you can. Either use the stump killer product, or a broad spectrum herbicide (Roundup, Weed & Grass killer, Brush Killer A, etc)
Take the undiluted product, add equal amount of water, teaspoon dish detergent (aids in penetration). Use a disposable paint brush- paint leaves of growing vegetation, and do not be sloppy- stuff kills anything green. Pour mix into holes that you drilled.