It's not easy to remove a tree stump, short of hiring an expensive tree removal service to do the job with a back-hoe. Tree stumps can be blasted out with dynamite, but that can be dangerous and requires the services of a professional. The options (outside of buying commercial stump and root removal chemicals) are limited, but will work if you are patient.
Rot the stump
Cut the stump as close to the ground as you can. Use a wide bit to drill as many holes in the top of the stump as you can. Fill the holes with water and a fertilizer that is high in nitrogen (45-0-0 would be best). Soak the ground around the stump and cover it with a plastic sheet or tarp. Put bricks or rocks on the plastic sheet and cover it with organic mulch. Keep the mulch and tarp wet and weighed down. Periodically remove the tarp and apply more fertilizer to the top of the stump. The stump will slowly rot away.
Burn the stump
Cut the stump as low to the ground as you can. Bore a series of vertical holes in the stump at least 6 inches deep and 3 inches apart.
Mix 1 part lead acetate with 1 part manganese dichloride, 1 1/2 parts cupric chloride, and 4 1/2 parts of sodium dichromate. These come as powders. Mix them with water until they are as thick as molasses.
Pour 1/4 cup of the mixture into each hole, then add a few drops of liquid detergent. This will help the mixture penetrate the stump. You'll need about 2 lbs. of this mixture for a stump that's 1 foot wide. Cover it with a plastic sheet and weigh it down with bricks or rocks. Let it sit for two or three months. Use rubber gloves when mixing the chemicals.
In the late summer or early fall, it's time to burn the stump. For a small tree, punch holes for a draft near the bottom of a 5-gallon bucket. Build a charcoal fire on the stump and put the bucket upside down over the stump. If it is a large stump, build a brick enclosure around the stump to keep the fire from spreading. To speed up the burning, make a reflector of aluminum foil and a simple frame to reflect the heat back onto the stump.
Saltpeter (potassium nitrate) hastens oxidation. Commercial tree stump removers are usually about 98 percent potassium nitrate.
Cut the stump as low as possible, soak with water, pour the potassium nitrate crystals on top and cover with a plastic sheet. You'll have to keep adding and soaking. After a couple of years, you should be able to turn a 2-foot-wide stump into powder.
Potassium nitrate is a commonly used fertilizer. It is also used to cure meats and give cigarettes an even burn. It is legal to buy and a safe way to remove tree stumps, although in combination with other substances it can be turned into explosives.