What you need:
*webbing, or fabric that you really trust. Webbing is everywhere, though. You could even trim a little off of a strap off your bag. I pulled mine off of a suitcase i found in the trash
*bungee cord that isn't too long.
*bicycle with rack
*needle and high quality thread or floss
Step 1: Cut and sew on webbing
Make sure you sew the top two far enough apart so that it's appropriate with your rack. Racks will have cross pieces of metal, and where those intersect with the side piece is where you'll be attaching the bungee hooks. Make sure that the distance between the top and bottom webbings is slightly shorter than the distance between the top of your rack and the hook at the bottom
The other thing to keep in mind is, depending on the size of your bag, you will need to offset the bottom webbing to one side or the other to prevent your heel from kicking the bag. In my bag in the picture, the bottom webbing is actually still a little to close to the center, and I need to move it maybe an inch to the right, so that when i mount it on the left side of my bike I'm not kicking it.
Sew it good and strong and make sure you use a high quality synthetic thread. Cotton threads will rot in the rain. Floss would work well, as well. I use upholstery thread, which is very strong. You can steal it at any big box fabric store, or buy it from your local fabric store.
Things that would improve the straps that differ from the picture:
*sewing over more surface area, specifically with an X in a box style. you know what i mean. A box with an X in it.
*(i will probably be doing this one) riveting into a piece of thin metal or wood or something on the inside of the bag. This will also give the bag more of a frame, if you're worried about the bag getting into your wheels.
Step 2: Threading in the bungee
Step 3: Secure your straps and loose ends
You'll have to experiment. If you don't have compression straps on the side, you could make a clip that lets you just connect your backpack straps together. This would be useful as a chest strap when you're wearing the backpack like normal, as well. You could also use a strip of double sided velcro to tie things on the other side.
Step 4: Connect to your bike
Step 5: Carry stuff
I should mention that my bag had some foam that gives it a little shape and keeps it from getting into the wheel (so far). Putting something to stiffen the back may help a lot. You could even just include a piece of cardboard and pack it in if you go on a trip or something. Or have a stiff something that you velcro in. Or rivet in with the strapping. Or....
Let me know if i did a crappy job or if you have any questions
and I might as well plug my photography website, while i have you here: GregClarkePhotography.com