Hi Daniel,
If you used the iFixit Battery Replacement guide for this phone, then you may have seen the information in the preface talking about the need to calibrate a new battery in order to get accurate battery charge readings.
So my question is to ask whether or not you did the calibration procedure as outlined in the guide.
[[Battery Calibration|Battery Calibration - iFixit|new_window=true]]
If you don't calibrate, then the phone will still be working off the data from the old battery and will miscalculate how much charge the battery has at any given time. Calibration resets that information so that it's correct for the current state of the battery, which of course has now changed with the new battery in place.
So yeah, charge it as high as it'll go, use it until it dies, then charge it again. Won't hurt at all to do that a couple of times more even; that'll ensure the phone has the most accurate data on the battery that's possible.