My iPhone 5s has needed a new battery for a while -- after all, it's 6 years old. Recently, however, I noticed that the display was bulging out of the frame and suspected that the battery was swollen. I took it to the Apple store to see if they thought that it was just the battery or it needed a new display. They said probably just the battery but there was no way they would touch it for safety reasons.
Despite some misgivings about whether the display glass and frame were separating and whether the battery could be removed safely, my son and I set about the repair using the repair guide and the battery kit from iFixit. I could not be more pleased with the results. Not only is the battery like new, the display went back in the case just fine.
1) Cable ties to avoid pulling on the home button cable
2) Removing display mostly by prying after gentle pull with suction cup
3) Lots of heat and careful prying (plastic tools only) on the battery
4) Great care placing the replacement adhesive strips
5) Use tool tips without handle (and magnet) to start screws during reassembly