**Disclaimer:  I am not a professional.  This is the first time I've upholstered a chair so forgive me for my mistakes.
Cut the new fabric
 Cut the new fabric using the old pieces as the pattern.  Cut a few inches larger all around so you have room to pull the fabric when you staple it to the chair.  You can cut the excess off later.
  I used about 5 yards of fabric for my chair.
For the seat deck, I used a large piece of seat deck denim and sewed a piece of the regular fabric onto it for the part that you can see by your legs.  This saved some yardage (i.e. money - $1.50 a yard for the seat deck denim vs. $ a yard for the upholstery fabric) on the upholstery fabric.  The deck denim also seems like a tighter woven fabric that must hold up to wear a little better but I really don't know for sure.  That's just my guess.
Attach new upholstery
Step 1:  Seat

I left the original material over the springs because it was in good condition.  Staple the edge roll on the front of the chair.  Cut bonded Dacron - I did 2 layers just cause one layer felt a little flimsy.

Center the fabric on the chair, smooth it out, pull taught and staple.  I stapled the front, back then the sides.  I worked from the center towards the edges to make sure there weren't any gaps or wrinkles in the fabric.
Step 2:  Inside sides

I used a layer of bamboo batting and two layers of Dacron.

I stapled the Dacron in a few spots to help hold it while I was laying out the fabric.

I really struggled with laying the arm fabric out and getting it folded, stretched evenly and stapled down.  I could have used a couple extra hands.

I need to stitch that vertical flap of fabric down.

Here is a side view of things to this point.
Step 3:  Inside back

Staple burlap to the back as a protective layer between the springs and the rest of the upholstery.
I wanted my back cushion to be part of the chair.  I didn't take many pictures of this step but I'll explain what I did.  I took a piece of muslin and put it over the burlap, stapling a couple times at the bottom only.  I stuffed it with polyester pillow stuffing to the approximate thickness of the old cushion.  I put one staple in the top of the muslin to the back of the chair so I could use both hands and adjust the stuffing.  I added more, moved it around and flattened it until it looked nice and was comfortable for sitting.  I put the seat cushion on at this point to make sure the back wasn't so full that the seat didn't fit.  Once everything was perfect, I finished stapling the muslin.

I put a layer of Dacron over the back to give a little extra softness.  The yellow part in this picture is the muslin filled with poly fiber.  I pulled the Dacron down to get a picture.

The Dacron smoothed out with the seat cushion in place to make sure everything would fit nicely.

I attached the back fabric and stapled everything down.  I'm hoping the seat cushion smooths out for a better fit with the chair as I use it.  I think it's all so plump right now that it's puffing up.
Supplies needed:
I bought everything from 
Diyupholsterysupply.com except as noted below.
- Upholstery fabric.  If you're in the Minneapolis/St. Paul area, I highly recommend S.R. Harris.  They have more fabric than you could ever imagine and it's all half off.  I got really nice 100% wool fabric for $10 a yard.
 - Thread
 - Seat deck denim
 - Edge roll
 - Stapler and staples.  I used a hand powered stapler and wow, my hand is sore.
 - Dacron (from Rochford Supply in Minneapolis)
 - Batting - I used bamboo because it was cheaper and more eco-friendly than cotton (from Hancock fabrics)
 - Burlap
 - Muslin (I used this to make my back cushion)
 - Ply Grip/Curve Ease
 - Cardboard tack strip
 - Metal tack strip
 - Filling for the back cushion (I got a 20" x 20" pillow from S.R. Harris pulled the filling out)
 - Foam for the seat (from Rochford Supply in Minneapolis)
 
I found the videos on 
DIY Upholstery Supply incredibly helpful.  I also checked a few books out of the library but none of them were exceedingly helpful.  The best one was 
Singer Upholstery Basics Plus by Steve Cone.