The most difficult part for me with the iPhone 5 was prying the battery loose from the underlying adhesive tape: slowly and carefully are key. Don't apply so much force that the case edges get distorted. Once the old battery was removed there were pieces of tape left stuck to the phone base. I removed some of this but left the pieces that were flattened out completely so that 1) the new battery would be at the same height inside the case as the old and 2) so it would stick in place like the old one.

Another tip: don't remove the display cable, just be careful not to tilt the display up beyond 90 °. Overall it was well worth the ~$50 saving between the $25 Amazon battery kit I bought and the iStore repair and iExperience.