There are a few approaches to removing Ivy. The first is to use a herbicide such as Glyphosate. It is most effective if applied to the base of the plant and slashing the vines will allow the poison to penetrate faster, quickly removing them.
For a poison-free approach, you can lay heavy black plastic or several layers of newspaper over the vine, blocking sunlight. After some weeks, you'll find the vine will have died off. You can then remove the cover and replant in the area.
Finally, the most labour intensive of the options is to remove the vine by hand. Start around the perimeter and pull the vines out of the ground. Try to make sure you remove all traces of the roots as they tend to grow back. A good quality pair of gardening gloves and a spade is a must.
Remember, although Ivy can look fabulous on masonry walls, it does come at a cost. This could be in the form of early mortar degradation between bricks or to such extremes as causing the wall to collapse under the vines additional weight. - MitchellMc
I've always cut and dug it out, and it's not as hard as it seems. I use a pair of loppers for thicker parts, even the roots, and a pruning saw to cut the ivy if it's growing up a tree.
My days of sheer braun are long gone, so for attacking the roots I trowel around the base of the trunk to find the roots. Then expose about 150mm and dig underneath the root. Depending on the size, I'll either use the loppers, or a reciprocating saw with a coarse blade. I haven't got one, but a survival saw could be handy. To lever out heavy roots, I recommend The Prong Company.
Success rate is 100%. By the way, removing the roots is best done when the soil is moist. And when done, put all the cuttings straight into your green wheelie bin. - Andy_Mann
I had a "little" problem some years ago. I used my true trusted method - salted water:
1. Put the salt directly onto the actual spot/roots - then pour boiling water onto it.
2. Salt the water (hot or cold) and carefully pour the solution onto the spot/roots.
Use this for weeds too. No harm to our "garden" pets, frogs, etc. - BibbyKat