One way to kill the root system on a tree stump is to rot out the stump. The rot will travel down into the roots and eventually destroy them. You can use chemicals to complete a tree stump rotting or you can use powdered milk, which you can get from just about any grocery store. Powdered milk will rot the stump and give you a lesson on growing fungus --- an interesting byproduct of using milk to rot a stump.
Drill as many 1-inch holes through the top of the tree stump as you can. Drill down as deep as the drill will allow.
Hose off the tree stump as well as the surrounding area, using a garden hose. If you cannot reach the area with a garden hose, fill a 5-gallon bucket with fresh water and pour the water onto the stump and surrounding area.
Pack the holes with powdered milk. You can forgo the powered milk and use a nitrogen-rich fertiliser if you choose. The use of milk is an organic alternative to using chemicals such as nitrogen fertiliser.
Place a plastic tarp over the stump, put mulch over the plastic and wet the mulch with the hose or bucket.
Remove the mulch and tarp once per week. Wet the area again and repack it with powdered milk. The stump will start to grow fungus. This is an indication that the milk is working. You can stop treating the stump once the stump crumbles in your hand. This is an indication that the root system is dead.