It sounds like several people, including myself, had trouble charging the new replacement battery past about 10%. This ended up being due to the new battery not having a non-conductive insulator. The way to solve this is by adding a small piece of electrical tape to cover the battery connector before screwing in the battery connector bracket over it. You may want to do this on the first go-around.
If you do experience that problem, I found the answer here and it worked perfectly: iPhone battery not charging after DIY battery replacement. Why?