To connect a serial Eeprom to PIC take a look to microchip web page, in the apps there are some applications to use serial eeproms like 24cxx and 93cxx.
To connect PIC to PC via serial port, you must use a driver to RS-232 level. See the MAX232 chip or any equivalent that converts 0-5v levels to -12 to 12V levels.
In the microchip apps there are too, the way to do the UART, or the way to make the serial comunnication, at differents bauds rates.
If PIC is 16C65, 16C67, 16F877, or more they have an UART and it's easiest to use.
Hope it helps, good luck :arrow: