You need to get back to de basics on networking, the way I understand you question is: "I have a serial port and I want to use is an Ethernet link". Sorry to crush your dreams but you don't have the real hardware to do so, I'll elaborate on it.
A serial connection is a physical connection that requires 3 wires (at least) tx, rx and ground. On the logical side you have an IC that coverts binary data into signals that are represented by discrete voltage ranges.
Ethernet is a layer 2 protocol, the layer 1 is provided by the technology used to transmit the signals (coax, up, fiber etc.) As you might see by now, you need a different set of hardware to convert the logical Ethernet frames into a stream of digital numbers, in fact this is call framing.
Since Ethernet has been an easy to use protocol it has been implemented as e preferred protocol for many network operators, of course one of the biggest is PPPoE where you have a PPP session over an Ethernet link. Of course this won't work with your example neither since you're trying the opposite.
If you're just learning and have all the time in the world you can attempt to write your own Ethernet framer over serial lines. This means you need to implement IEEE802.3 into the driver and then you need to serialize the data to push it as a stream of bits over the serial line. Of course note the following drawbacks:
- Your driver won't be able to fully support Ethernet, you need some support at hardware level to implement some signaling (example, auto negotiation, CSMA/CD, etc)
- You driver will be pretty much useless unless you back in time where 115.2kbps is top speed in data transfers
IMHO there are more exciting projects that you can pick up in the networking field for device drivers. You can for example attempt to buy a NIC and develop the device driver for it from scratch and you can optimize certain areas. Finally, remember that most of the Ethernet implementations are now done in hardware so you don't have to do anything but filling a few registers on the MAC and voila!