What to do after replacing the phone's battery?
You shouldn't need to do anything, the time remaining displayed in the phone may be incorrect for a short time (a few days at most), but it should correct itself with a few charge cycles. Wiping batterystats.bin has no effect on the actual life of the battery, but it can have an effect on the display of the remaining life of the battery. Source: XDA/Google

If your phone is getting hot, it is an indication something is not right... and since you replaced the battery that probably is not, and likely was not, the real issue. Either you have a hardware problem, or a software problem, the heating is likely due to the an application not sleeping properly and keeping the device awake, which will also reduce you battery life. I would install GSAM or BetterBatteryStats (paid), and possibly WakeLock Detector (root needed), and try to figure out exactly what is using your battery up. Although more rare, this could be caused by defective hardware as well.