Reupholster Thin/Small Metal Vinyl Chairs  

I can't tell from that photo what the seat deck consists of or how the fabric is finished underneath, but I can tell you that you'll need to know how to do a boxed cushion cover as far as the fabric goes. This is a very basic upholstery technique though it can be tricky as you need to measure and cut very accurately and then make sure the layers match up exactly before sewing You don't need to do welt if you don't want to but you should at least topstitch the seam allowances down if you want a nice finished look.

Make sure you get a fabric that will fit under the presser foot of your machine when layered. You will have four layers at the corners and a lot of upholstery fabric is too bulky for the average home sewing machine.

As for the backs, they probably have machine stitched seams. You can try carefully undoing the seams and using the pieces as patterns for the new fabrics; however, the new fabric will probably not have the same degree of give to it as whatever is on there now and your seams are fairly likely to look rippled when you pull the cover onto the frame. I have redone chairs like these and I find it easier to hand stitch the entire seam on the frame, but that requires are lot of careful pin fitting first. I think it's worth it, though. But it's not a beginner technique. (Neither is trying to do it by machine unless you have exactly the right type of fabric, whatever that might be.)

The back covers most likely need to be hand stitched closed along the bottom but again, hard to say without a closeup of how it was done originally.

You are looking at a lot of work for the level of quality of those chairs.