Why don't you buy a ethernet card for the system? They are cheap. Open the cover, look for an open slot. If you are not sure of the bus type, you can have, in a browser, go to wikipedia, type 'pci bus' for example and they will show you a picture of the connector. Most systems built in the past 10 years have PCI bus. If this is a laptop, you can buy a USB bus card.
To build a serial cable, you can do it, but, you need to have soldering skills, and know how to count the pins. Much more difficult than adding a card.