How to Choose a Battery
Don't bother trying to make your own.  There are several models of external chargers with higher capacities than you found.  For example, the Anker® Astro Pro2 is rated at 20Ah, is designed to charge the Galaxy S4 among several other models, and is available from Amazon for $100 (marked down from a $300 list price). It can also be used for laptops and tablets.   I doubt if you can build your own with that capacity for $100.