My iPhone 5 was glitching and not charging well, so I tried restarting it. Unfortunately it wouldn't turn back on, even after a hard reset. After determining it was probably a battery issue, I turned to the internet for help! I could've let Apple replace it (for ~$70), or I could order the whole replacement kit from ifixit for ~$20. Guess what I chose :-)
The repair was fairly straightforward. The hardest aspect was my own ignorance, never having played with computer/phone hardware before. I was afraid I might accidentally break something, but by following the steps my phone is now working better than ever. I don't remember the battery life ever being this good!
I marked out and labeled spaces on the paper tray that came with the kit to separate all the tiny screws that come out during the disassembly. This made reassembly easy.
Getting the battery out was the most difficult thing because it was sticking too well to the adhesive sticker (which would not come off even after warming it with a hairdryer). Using the plastic wedge tool worked best. It took a fair amount of wiggling and prying.
Don't forget: lithium batteries can't go in the normal trash! Put the old battery in a plastic bag and take to a local electronics store (ex. Staples). Most places will have a special container with which to collect this type of battery for proper disposal.