The best battery case for the iPhone 5S is the Lenmar Meidian Battery Case. It has a fast charging battery that doubles the life of your iPhone 5 or iPhone 5S.
Only 0.6 inches thick, the Lenmar Meidian Battery Case has a 2300mAh lithium-ion battery inside. This is enough to charge a dead iPhone battery to 100% in 2.5 hours.
I’ve been testing it for a year — and it’s a reliable power source. Once it is attached to the phone it is essentially maintenance-free. Charging is via a standard micro-USB cable.
The Lenmar Meidian Battery Case is available at Amazon for about $25.
Another Option for iPhone 5 & 5S
The Morphie Juice Pack Plus is another good charger I’ve tested. Functionally it is similar to the Lenmar case, but the retail price is higher (around $40). It has a battery capacity of 2100mAh — a bit less than the Lenmar’s 2300mAh battery. It will charge a dead iPhone 5 to about 88%.