Before you can cover the chair with your new fabric, you've got to get rid of the old fabric. Turn your chair over, looking for the seams. Fabric is often attached to chairs using a staple gun. Either cut the fabric off the chair to expose the filling, or pry off the staples holding the fabric in place.
You'll need a new fabric for the chair, but if the chair is old or uncomfortable, you might choose to upgrade the padding. A staple gun with staples can secure the fabric. Plan to get a couple extra yards of fabric so you have enough.
Once you've removed the old fabric, cut a large swatch of new fabric to cover the chair. Remove the old padding and put yours in its place. Cover the padding with the fabric. Staple one side, then pull the fabric taut over the chair. Staple the other side. Stretch and staple all sides in this manner.
It's easier to reupholster a simple chair, like a dining room chair that has a padded seat, than it is to reupholster an arm chair. If you're new to furniture restoration, start with a small project.