First off, when working the property, be aware that poison ivy can grow as a ground cover, as a vine or in a bush form that can make it hard to recognize. The hairy vine all by itself, in addition to the foliage, can cause a reaction in sensitized individuals.
To kill it you can either hand-pull it (wearing adequate protective gear) or very carefully use an herbicide such as glyphosate according to the label instructions. (Glyphosate is an indiscriminate herbicide and will damage all green plants, not just the poison ivy, so read the instructions and be careful if you are working around desirable plants.) Systemic herbicides can be brushed onto the poison ivy to control it around your other precious plants and lawn. You can also “smother” it by cutting it off close to the ground and then covering it with several layers of newspaper or cardboard followed by a thick layer of mulch. Any one of these approaches may need to be repeated at least once.
Unfortunately, this is one persistent plant. If you rip out the plant, never burn the debris as the smoke will carry the irritant. Also, the plant is deciduous and loses its leaves in the winter, but sensitized individuals may still react to the leafless vine.