I have prepared an illustration of how to create seams and how to place skirt panels (see below). It is very difficult to describe just how to do this, so the illustration is a must.
The seat cover itself is a really simple piece of work. The springs are 52-1/4 inches wide. With adding about 1 inch to this length to make sure the seams are on the sides of the foam (as opposed to being sat on) and another 1-1/2 inches for seam flaps, the springs are too wide to work with the typical 54" wide vinyl, so plan on making length-wise cuts. I can't say exactly how much to buy, as I bought remnants at a discount, and never really figured it up.
The "skirt", the part of the cover which goes from the top of the seat or back to the frame, needs to be 10" wide. There are three sections of skirt for the seat and three for the back. The front section is 46", and the two side/back (they wrap around and meet in the middle of the back of the seat, or the bottom of the seat back) should be cut about 55" so you have plenty to work with. The 55" measurement should be verified, since I just cut my pieces out of leftovers, and then trimmed the ends as they came together in the back.
The seat/back section should be cut by laying the springs on the vinyl and tracing around them with a marker. Hold the marker such that it adds close to 1/2" to the overall outline. The springs will not be perfectly straight along the long measurements, so use a straight-edge to straighten out these lines. This is the line that you will sew on. Make another outline 3/4" from the original. This is the line you will cut on. If you do everything right you will have a 3/4" flap (see illustration above) extending past the seam.
First sew the sections of skirt together, end-to-end as shown in the illustration. The three pieces combine into one long 10" wide piece with decorative seams to be seen in the front of the seat, or the top of the back The shorter piece is not as wide as the seat, so its seams are inset from the corners a few inches on either side. The two remaining ends are NOT to be sewn together at this time. To attach the skirt to the seat/back I stapled it first (I am SURE this is not the way a pro would do it, it was just an easy way for me) with a small paper stapler. This way you can position the front panel with its seams evenly spaced from the corners of the top/back, and it won't slip or stretch. I put a staple about every three inches and about 1/4" from the edge (remember you should be actually sewing about 3/4" from the edge). Just lay the skirt on the top/back face to face, line up the edges, and start stapling. I did not carry the staples around the corners, just along the front to keep my alignment. The corners are the hard part because they are round, and the skirt is a straight piece of material. You have to get rough with it sometimes (this is where practice makes perfect seams) but you can get it to work, just go a little at a time. The skirt will buckle, and this is actually what you want. You might want to make a removable mark on the vinyl at the center of the backside (or bottom if referring to seat back). This is so you know to stop sewing when you get to that point. This where I cheated a little for the sake of ease. I sewed around the seat in opposite directions starting from the front. When I got around to the back, I just let the skirt overlap itself, and trimmed it to fit. Then I just sewed the skirt to itself across the 10" dimension for added strength. Normally a nice seam is made in the skirt here, but this requires more precision than I was capable of, and this seam is not seen when the seat is installed. The same goes for the back. When you are done, and unfold everything the skirt should hang at a 90 degree angle to the top/back. Be sure and remove the staples.
At this point you could actually install the new cover, but it would not look "factory". To give it this look you must fold the flap of material that is leftover from the seam and top stitch it. The illustration makes this far easier to understand. This decorative seam also reinforces the whole thing.
Installation of the cover: If you can get your foam cut for you by a pro, by all means do it this way. Explain that straight lines are imperative, as the best sewing in the world will be screwed up by crooked foam. If you are going to cut it yourself, the best bet is to go to the thrift store and buy an electric carving knife. They will cut foam like butter, especially if you spray a little Pam on the blades first. For the seat, I used 2" foam of the higher quality. The higher quality foam is usually some other color than blonde. Mine is purple, and is firmer than the blonde stuff. The seller told me it is what the pro's use, and will last longer. I used 1" on the back. I did find a use for the hog rings. I used them to hold a piece of carpet to the springs as a backing for the foam (pile side up). I got my carpet from the dumpster behind the carpet store. It was brand new carpet left-over from an installtion. It is imperative that you put something in-between the foam and the springs as they will cut into the foam. I just cut the carpet about 2" bigger than the springs and force formed it around the edge wire. There are some wire struts that you can hog ring it to. It also keeps the edge wire of the springs from cutting your vinyl and makes the seat much firmer. Do the same for the back. Place the foam on top of the carpet, then slip the cover on. I lined up the front and side seams first. I figured if the back seam was not exactly perfectly aligned, no one would know. I set the seat on my bench with the front edge hanging out in space. I was then able to mash the clips (see illustration) into the channel while verifying the the front seam was even. The seams should not be seen from the top of the seat (you should not sit on them). This is why you make the top a little bigger than the actual dimensions of the springs. Place clips as often as you can along the front edge of the seat (in between each structural seat brace). I used some cheap pliers whose jaws I ground smooth with my bench grinder to pinch them into the channel. This is important! To keep from ripping the vinyl as you squeeze the clips into the channel, do not attempt to stretch the vinyl with the clip. Make sure that there is sufficient vinyl poked into the channel before applying the clip. With the front clipped into place, I flipped the seat upside-down and began gently stretching it along the back side. This is where you are most likely to puncture the vinyl with a clip. I got mine nice and tight, although I did have to re-do a couple of clips as I went. The bottom skirt will just fit. The back will have a little extra, considering you only use 1" foam.
I think this setup leaves you with a little taller seat, as I found it a somewhat difficult to wedge the seat bottom under the back. Use a little furniture polish on the surface of the vinyl, and this will go much easier. The seat is also very flat, which I believe differs from a factory seat, in that they have a little hump in them (I think - I have never actually seen a factory seat in good condition).
Let me know if you have any other questions.