This was my problem this morning. Battery never even entered my head. I noticed this bulge, same location near the volume buttons, screen had a slight separation with light emitting from it. Apple store rep looked at it and I'd have to make an appointment, they would have to look it... mentioned "probably needs a new screen, about $250 right now..you never dropped it?" - I dropped it just once when I first bought it in 2012..it was in such a strong case I did no damage or scratch it, no contact with water. I figured I would just wait until the new 6 comes out, phone still worked.
Friend mentioned this by this place called UBreakItIFixIt, I look it up on the net..some mixed reviews, some happy, some very unhappy people/experiences. A screen replacement would have been 120 bucks. For the heck of it I stop in, show it to the guy..quickly mentioned the "battery is going to blow." That caught me way off guard. He opened right there and showed what it looked like..black thin battery was slightly inflated like a small balloon. Showed me what the normal battery looks like. WTH!?
Anyway, 69.99 to replace it, took no more than 3, 4 minutes. Now what kind of actual replacement was put in my phone..3rd party battery or what? I can't be 100% sure. Whatever it is, it's ok for now - heck of a lot better than an acid filled timebomb. The phone works though, bulge is gone. Big piece of mind now thinking this was a screen problem when it was actually a dangerous problem. This is a first for me, I had iPhone 1 from 2007-2010, then iPhone 4 from late 2010-2012, never had issues with the internal batteries.