Have the tree inspected by an arborist before removing it on your own, especially if it is large or fallen due to storm damage. Large trees are best removed with heavy equipment and service from a tree removal company. If the tree is manageably small, you may be able to uproot it or cut it down without assistance.
Ensure there is a clearing in all directions where the tree can land if it happens to fall from its leaning state.
Dig around the root system of the tree, if it is small enough to manage this and if doing so will not harm the neighboring tree’s roots. Use the shovel to dig beneath the root system and pull it up, or use two shovels to pry the tree out of the soil.
Cut the tree as close to its base as possible with an axe or chainsaw, if you choose not to dig out the root system. Make sure you have an escape route planned in case it falls in your direction. If the tree is especially large, heavy equipment by professionals may be required. Cut the tree it into smaller pieces for disposal, if necessary.
Grind the remaining stump down with a stump grinder, which can grind it down to 6 to 12 inches below the soil surface.