Q: Is there any type of chemical that you can feed a tree that will kill it and the roots? Also, How long would it take?

A: I would just remove the tree, without killing it first.

A live tree is usually safer to remove than a dead one as it has a more predictable "consistency" to it. Dead trees can have hidden weak spots in them that make them less predictable and much more dangerous to remove.

As for the roots, they will generally die off after the tree has been cut down.

You can also have the stump ground out to clear the area for re-use. Should any roots begin to sprout up, treat them with a non-selective herbicide (be careful of other plants or grass, as non-selectives will kill them, too) and that should take care of them.