In order to increase your backpack size you need to complete the Claptrap Rescue missions in playthrough 1. Each rescued claptrap gives you a backpack sdu which increases your backpack size by 3. In total you can increase your backpack size from 12 to 42 if you do all the playthrough 1 rescue missions. You can then go through playthrough 2 and there is a chance that the first 5 claptraps will give you backpack sdu's. If you get all 5 there, you'll have 57 slots. You can then go to the DLC and gain an additional SDU from each playthrough. This would give you a maximum of 63 slots. However, most people have approximately 42-50.
Locations to all clapraps which need to be rescued: http://borderlands.wikia.com/wiki/Claptrap_Rescue_Missions