Connecting the MFRC522 RFID Reader/Writer to PC
There's an existing Arduino library that uses the module's UART interface, rather than the more common SPI. It provides basic methods for reading the UID of a card and R/W operations on blocks. You could easily port that library to your favourite PC programming language, connect the module to your PC probably through some USB-UART adapter and get to work. 

If your PC has an actual TIA-232 serial port and you want to use that to talk to the module, you'll need a driver like the MAX232 to sit between the module's TTL UART and the PC serial port.

Use this as a guide:



All GNDs must be connected together.