The easiest way to carry a backpack is on your back. If you're looking for a bag for cycling, then you either need to invest in some pannier bags, or some bikepacks:
I have a 20L Revelate seat bag on my bike and I love it, I still carry a small backpack, but I try to put all the bulky items in my seat bag. Heavy stuff should go inside a frame bag, but those are harder to come by because they need to be custom fit, but if you have a mainstream mountain touring bike like a Salsa or Surly, etc., then you can get one stock that fits your frame.
When riding long distances, you want a little weight in your seat as possible, more weight on your back means more weight on your prostate or other tender bits. I'm still in the process of building a tour divide bike like the one pictured above, after all is said and done I plan on having mostly nothing on my back except a 3L water bladder and some empty space in a backpack that I only plan on using during those long stretches during the tour divide where you're without water and services for several days in a row.
Bikepacks on a full suspension bike:
