To start streaming data from your Arduino over a Ethernet cable, simply

1. Upload the tiny server sketch to your Arduino
2. Plug in your ethernet shield, connect the Arduino to your computer with an ethernet cable and wait about 30 seconds for the Arduino server to boot up
3. Click the "Start Reading" button below

Start Reading

Pin A0's value: —
Pin A5's value: —

See the GitHub repository at https://github.com/sfentress/ArduinoEthernetCom.
To read more about streaming data to your computer over an Ethernet cable, read this blog post.