In addition to solution 1 you can also use a PC I2C interface. These are quite cheap and have an USB interface which is usually accessed as virtual COM port. Such interfaces should provide an SDK with a library which can be also used with C# when using PInvoke.
If you just want to read / write to the EEPROM without using your own code, you can use PonyProg but you need the I2C interface hardware as described at
PonyProg - Serial device programmer[
^] which requires a parallel LPT port which is no longer present with recent PC systems.