When am I able to upgrade my inventory and ammo capacity?
Solution 1:
Ammo and backpack upgrades are acquired on the black market. These purchases have to be made with Eridium; a sought after mineral on Pandora. Once you reach the city of 'Sanctuary', one of the first missions you encounter is entitled 'Plan B'. You will receive 4 Eridium from Scooter, allowing you to make your first purchase at 'Earl's black Market'.
http://borderlands.wikia.com/wiki/Eridium
Solution 2:
You can upgrade through Crazy Earl with Eriduim. I am currently maxed out at a 27 space backpack. The upgrade costs are set at 4, 8, 12, 16, and 20.