Locate most tree roots within the first 6 to 24 inches of the soil. Some trees, such as oak trees, have roots that burrow deeply into the soil, but usually not farther than 3 to 7 feet. With root removal comes stump removal and most removal occurs by working through the stump first.
Use a stump removal chemical on the stump, which then travels down through the roots. Drill holes in the stump through which you will pour the chemical. Wait for the stump and its roots to decompose. Break up the roots and stump with an axe when they turn into a spongy texture.
Burn the roots away by starting a fire on top of the stump. Gather smaller pieces of scrap wood and brush to place on top of the stump to heat the fire up. Keep adding to the pile to allow the fire to burn long enough to burn through the stump to the roots. Surround the stump with some chicken wire to contain it and protect children and animals.
For quick but labor-intensive removal, pull up tree roots by use of a backhoe or skidsteer or grind up the roots starting from the stump with a stump grinder. Borrow either piece of machinery or rent from a hardware store. Stump grinders work to about 6 to 12 inches below the ground. Skidsteers or backhoes allow you to use their teeth to dig below the surface and pop out the roots. Read the operator's manual before operating machinery and wear protective gear such as goggles.