Get a rack and strap your backpack to it. That's what I do. You can also look into cycling specific backpacks but they can get pricey. You can also look into panniers that double as backpacks off the bike.

Having a sweaty back is a result of many factors. If its hot and/or humid you will sweat. If you are exerting yourself you will sweat. If there is little to no ventilation between pack and rack then sweat will build up. I have a Jansport Revolt 32 I bought from a discount store for about $30. It has an egg crate design on the back to help increase air flow (among other things). I still strap it to my rack though.