If you want to remove the roots (you didn't say how big the tree was) dont chop it right down, leave about 1.5 feet so that you can rock it and disturb the roots, having dug around them. I have successfully got rid of quite large stumps by cutting or slicing at the thinner parts of the roots. It took several attempts but I eventually managed it.
If its too big for that,cut it down, bore big holes all over it and put in plenty of Brushwood Killer. I've forgotten the brand name but any garden centre will know it. Some people leave enough stump to grow clematis or similar up over it. Good luck anyway.