The following diagram may be helpful to asses what additional hardware components you might require to establish communication between your UNO and a remote computer using Software Serial Port (UART). The diagram also has depicted the detailed (tested) circuital connections of the MAX232 chip which contains built-in 5V/PM10V converter.

Procedure:
1. Power down the UNO Board.
2. Place the MAX232 chip on the breadboard.
3. Complete the circuit around it (MAX232) placing the 1uF capacitors (the polarity must be correct!).
4. Connect 5V and GND of UNO with the corresponding Vcc and GND pins of MAX232. Do not connect the STX and SRX (DPin-6, DPin-7) with the MAX232.
5. Apply power to the UNO.
6. Use DVM and measure DC voltages at Pin-2; the reading should be around +10V (+6V to +10V acceptable). Measure voltage at Pin-6 of the MAX232; the readng should be around -10V (-6V to -10V acceptable).
7. Power down the UNO. Short Pin-12 of MAX 232 with Dpin-6 of UNO. Short Pin-11 of MAX232 with DPin-7 of UNO.
8. Get the RS232 to USB Converter Cable and take the 9-pin side. Now. short the following pins together (difficult job!):
Pin-5 with GND pin of MAX232 and UNO.
Pin-4, 6, 1.
Pin-7, 8.
Pin-2 with Pin-14 of MAX232.
Pin-3 with Pin-13 of MAX232.
9. Connect the USB side of the converter cable with any available USB port of your second computer. (Not the Computer with which your UNO is connected.)
10. Install the device driver of the converter cable and check that the PC has recognised the cable at some COMX port (Virtual COM Port).
Now, your turn to write the communication software! Post your codes so that we can participate in the development.
Good Luck!
BTW: Once you have finished the hardware, you need two software routines for testing the network:
(1) For PC1 rut-1.ino. This program will receive a command/character (say 5) from PC2; the charcater will be shown on Serial Monitor of PC1. This program will also allow you to send command to PC2 via Serial Monitor. Hopefully, you can develop this program. See examples of the IDE.
(2) For PC2, rut-2 program. If you like, you can use this GUI Program of Baud Rate:4800. From this GUI Program, you click on the menu Click to Roll Call; in response, character 5 will appear on the Serial Monitor of PC1. From the Serial Monitor of PC1, you enter 6 (and then click on send button); in response this message (MicroTalk-8085/80c86/8051 is Ready…!) will appear on the Input/Output Box of rut-2 of PC2.