Hello! I would like to simply send a file from my pc to the microSD card i have on my microSD shield using the serial connection.
You can't do this directly. You can send data from the PC to the serial port. The Arduino can read serial data. The Arduino can open a file on the SD card. The Arduino can write data to the SD card. The Arduino can close the file on the SD card.
There are examples for reading serial data. There are examples for writing to the SD card.
It is trivial to combine them.
What (Arduino related) part(s) are you having trouble with?