You cannot connect the EEPROM directly to the com port. You will need a microcontroller for example to receive the data from the com port and then write it to eeprom.
You could use handshaking lines of the comport to directly program an eeprom but you will have to write PC software for it .
Very easy, one of my RS232 tutorials does just that - except it doesn't store the text in memory, there's no need - the LCD module keeps displaying what it was given until power is turned off (or you overwrite it).
If you want to store the text in non-volatile memory it would easy to add, most PIC's have a small amount of EEPROM data memory you can use.