Before you can upload a program via serial, you need to have a bootloader on the mcu and have the fuses correctly set. You can either buy an mcu with the bootloader already installed, or use the mcu removed from a working Arduino, or you can program a bootloader on to a blank mcu using an ICSP (in-circuit serial programmer). You can use an Arduino as an ICSP.
First thing to do is a simple loop-back test of the RS-232 module to see if it's functioning. Disconnect it from your arduino pins and jumper the TX and RX pins on the module. Open the arduino IDE, select com port 13, and then open the serial monitor on the IDE. Anything you type on the serial monitor and send should echo back the same. If that works then it's probably just a naming convention that is tripping you up. Sometime a serial converter modules names it's signals from the users point of view not it's own point of view.
This of course assumes you have a bootloader on your AVR chip, and that you know how to properly time a manual reset button on your breadboard when you attempt to upload to it. If not ask how.
The picture is small and hard to see but assuming your MCU has a bootloader and the fuses are set right, do you have a pull-up resistor connected between reset pin (1) and VCC. Do you use auto reset feature for uploading sketches? If so then do you have a small capacitor connected between reset pin and the RS232 DTR pin. If your not using auto reset then you need to press the reset button at just the right time to invoke the bootloader while uploading your sketch.
WHen first working with a device like the serial board you show it is convenient to have a program preloded in the 328 chip that constantly outputs to the serial port. That way when you hook it up you have a way to get the transmit and recieve wires sorted out. If you hook it up and the PC is able to recieve data you probably have things hooked up correctly.
