Interfacing with desktop computer console software
For debugging:

Configure the UART on the ATmega to output TTL serial. If you are using the Arduino libraries, then Serial is what you want. Connect the TX pin of the ATmega to the RX of the USB/Serial cable, and vice versa.

Open the serial device on your computer in the program of your choice. When you write to the UART on the ATmega, you should see the messages in your terminal.

Pretty straightforward.

For programming:

If you're going to be using C and flashing things manually, you'd best go and buy an AVR programmer so you can program the ATmega directly without messing about with a bootloader. A programmer is ~15USD, so you might as well buy one. Then you can use the GNU AVR toolchain, which is very easy to use. Compile code with avr-gcc, and program the device with avrdude.

If you really want to use programming over serial, you'll have to manually hit the reset button each time you program the chip, because (as Mikael pointed out), that serial cable lacks the lines necessary to automatically reset the chip (and drop to the bootloader).