Removing the root or stump is a different thing, the op asked how to kill it off completely.
The best way to get rid of a stump and roots is to call in a tree surgeon with a stump grinder, this will take them from ten minutes to a few hours depending on the size of the stump, I have ground out hundreds of roots and stumps with a stump grinder.
Cathfromsaron, that is exactly what I have done in my wild garden, I took down about 5o small trees, I just cut them down just below ground level and grassed over them. No problem.
But it rather depends on what the OP wants to do with this piece of land now the tree has gone.