After a year of increasingly constant use, my son's iPhone 5s battery began to lose power rapidly and/ or suddenly (sometimes going from 40% to 0% in a blink)! After looking up the problem at various sites, I decided to replace the phone's battery.
The iFixit repair guide was very straightforward and sufficiently detailed to get the job done in about 1/2 hour. The warnings and notes are especially helpful as they help to avoid inexperienced mistakes (such as not prying up the entire connector, just the connection itself)!
The toughest part was with the battery adhesive strips. Although I pulled the adhesive tabs as described, they ripped anyway, so I was very patient and careful to warm up the existing strips with a hairdryer and used an old credit card to gently pry out the old battery. Once it was out, I peeled the remaining strips and then reapplied the new strips that came with the repair kit. Voila, all good!
I do have experience with phone repairs, this being my 9th or 10th iPhone, but my first battery replacement. The iFixit guide is perfect for the iPhone 5's repair. Interestingly, when I opened this phone for the first time, 2 of the 4 screws holding the connector shield were loose or missing! They were either jarred loose or it was just not put together properly. Ensure that all the connectors snap appropriately when reconnecting and never force them. You should feel a satisfying click for every connector, no matter how small.
When initially opening, I find that a hard thin metal object, like a razor blade, is useful in starting the initial prying, then use guitar chips, or other opening tools to wedge the cover till it snaps open.