If the frame of the deck chair is glued together, remove the old sling by cutting the seams that form the casings with a seam ripper. Use the old sling as a pattern to cut the fabric for the new sling. The best choices for the chair fabric are weather-resistant canvas or nylon. Finish the edges of the sling by wrapping them with double-fold bias tape. Weather-resistant bias tape is available online. Choose contrasting bias tape to add an accent color to the chair. White bias tape on navy fabric will give the chair a nautical look. Use an upholstery needle and clear nylon thread to attach the sling to the chair.
Some deck chairs are held together with screws. You can disassemble the frame on these chairs by removing the screws that hold the bar at the top of the chair back and the bar at the front of the seat. You will only need to remove the screws on one side. Slip the sling off the bars and measure its length and width. Replacement slings are sold online, or you can make your own.
Older teak chairs often have a slot cut into the wood at the top of the chair's back and the front of the chair's seat. The sling for these chairs has a wooden slat attached to the ends of the fabric. Remove the fabric by slipping the slats out of the slots and pulling out the staples that hold the fabric to the slats. Finish the side edges of the new sling with a hem or bias tape. Staple the new fabric to the slats and replace the slats in the chair frame. If your old sling had stretched, cut the new sling a few inches shorter.
If your chair has an unusual sling or the fabric has become too stretched or damaged to use as a pattern, you can have a sling custom made. Ship your damaged sling to a company that specializes in reupholstering patio furniture or take it to an upholstery store.