My advice is to get as much off as you can when it is green and living. Then I used 'root out' to deal with stumps. It is a slow process but works - the stump will remain but will not resprout, maybe take a year. Break bark and apply rootout crystals and cover with some plastic to protect from rain wash off.
I killed an ivy root by cutting some slashes into it and pouring some waste oil on it. Worked a treat but you need to make sure the oil stays at the root and doesn't spread. I would put on a little at a time to prevent it spreading and harming other garden things.