The Arduino is currently using serial over USB, but what you want is RS232 (which is more of a communication voltage standard as applied to serial comms). The TTL-serial data from your Arduino needs to be converted to RS232 levels, which is something like +/- 9 to 12 volts. Usually, people use a MAX232 chip (or compatible, they're cheaper) which is designed for this exact purpose. It would require soldering up a PCB with some capacitors etc. But there are premade shields for this, Sparkfun has one: https://www.sparkfun.com/products/11958
However, you'd need to remove that shield in order to program the Arduino through USB. There may be other shields out there that allow you to switch the RS232 connection on and off, or let you tie to other pins and use SoftwareSerial instead of the hardware ports.
Once you have the electrical side of things figured out, the rest is easy...just use some the Arduino serial examples. A "Serial.println("get ip");" might be your start.