What to do after replacing the phone's battery? I have a Nexus 5 (1.5yo; I know, it's old but still worthy!) and the boy was heavily suffering from short battery discharges. I'm a phone heavy user so I figured out I had depleted the battery's life, and decided to buy a new one.

I used that great iFixIt guide to replace the battery and everything is fine, except that my phone keeps lasting 2 hours and something after a full charge, just like it did with the old battery.

Are there any special procedures I can take to ensure I can have a lasting battery life, or could this be related to the power port or other internals? I feel the back of the phone overheated just like it happened with the old battery as well.

UPDATE: it seems that battery I bought was faulty. After some weeks the phone dropped on the floor and the battery stopped working completely. The new one I got from the warranty gave the phone new life, as expected :)
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.