As Steven said, PICs don't have unique serial numbers built into them at manufacturing time. Let's be clear that we're talking about a number that would vary between two chips of the identical model.
Most PICs do have a chip ID encoded into them. This can be read by a PIC programmer to determine the type of PIC it is connected to. As far as I can tell, only the 12 bit core PICs don't have such a chip ID. All the other PIC can be identified electrically, although there are quite a number of differing ways they have to be put into programming mode and then the chip ID read out. PIC programmer software that can identify a arbitrary PIC is not trivial.
If you really meant serial number, then this has to be programmed into the part after production. We have done this many times. During the production test process when the firmware is programmed into the PIC, you set it up to also get a unique serial number and program that in too. One easy way is to start with a master HEX file as produced by the firmware tools. This contains a blank serial number. You write a small program that grabs a new serial number according to whatever your strategy is, reads the master HEX file, substitutes the new serial number for the blank one, and writes out a temporary HEX file. The temporary HEX file is used to program the part, then deleted.
For better robustness, you store the serial number state in EEPROM in the production jig, not in a disk file. The system asks the jig for a new number, and this jig is designed to never give out the same number twice. When it's assiged serial number range is exhausted, it refuses to work. This is especially useful if you have a remote manufacturer building units for you. This way if they mess up something on the computer and restore from backup, it won't also reset the serial number range. Yes, this has actually happened.
If the serial number is to be read from outside, then the user ID locations that most PICs have might be the appropriate place to store it. These differ from general program memory or EEPROM in that they can be read even when the chip is code protected.
If you get your PICs programmed by the Microchip or a distributor, you can usually get them serialized in some way. However, factory programming is only for mature high volume products. If you think you'll be upgrading the firmware regularly, then don't use factory programming. The lead times are long and you'll be stuck with old versions in the pipeline. Your device will have to be tested after it is built anyway, so adding a step to program the PIC is usually very little extra burden.