How to connect attiny to serial port so that I can use it with Arduino IDE
If you manage to burn the Arduino bootloader, than you should be able to upload sketches from the IDE similarly to the technique used for the Single Sided Arduino (through the serial interface).

This circuit reference could be useful:


Concretely the lower-left part of it, which shows the pinouts of the RS232 connector used along with some transistors to lower the potencial 12V to TTL logic. But if the RS232 to USB convertor shown in your photo is the "PremiumCord ku2-232a",


or a similar one based on the "FT232RL" chip, than it should output a maximum of 4.9V as HIGH on the RS232 side, according to it's datasheet. Therefore you should be able to safely use it without the transistors.

To the construction of the device: I would just tie the according pins on the ATtiny13 (datasheet): PIN5 on ATtiny13 - MOSI (Master Out Slave In) to PIN3 on the RS232, PIN6 - MISO to PIN2 and the PIN1 - RESET to PIN4. Than PIN4 on ATtiny (GND) to PIN5 on RS232.

But you still need to get 5V from somewhere and it needs to share the same ground with the RS232 connector (preferably from an USB of the same PC).