Since you have LEDs for RX and TX on your board, you can connect the board to the PC, select processor and port in the Arduino IDE, then open the serial monitor and set to 9600 baud locations. The RX / TX LEDs should flicker briefly. Then you can simply send any longer string of characters to the board and watch the LEDs. The RX-LED (your green LED) should flicker during the transmission, which indicates that the data has been received.