Learn how to program using the command line with AVRdude, or if you buy a programmer that is compatible with AVRStudio, use AVRStudio.
Learn what the fuses are for the tiny2313.
Wire the board with a power supply and a crystal (if you plan to use one)
Connect the programmer. (You can mess up badly on this...)
Check if the programmer communicates with the chip.
Set the fuses.
write a program in Arduino and compile.
get the .hex file for that program and download it with the programmer.
Test and report.
I'm quite sure you can find enough resources around the web about how to perform each of these steps. and I guess more people will complete the info and fill you in on what you can't find.
If you can find any hardware files for the ATtiny:
upload the 'arduino as ISP' file to your board,
select the hardware file for the ATtiny,
connect up the arduino pin 10 to the ATtiny slave reset, arduino 11 to ATtiny MOSI, arduino 12 to ATtiny MISO, and arduino 13 to SCK.
And then upload your sketch as normal. That is if you can find find some hardware files. Otherwise, you will need to do it the other way, as mentioned above.
To put in another way ... is there a way I can upload my own bootloader code, and not the Arduino bootlaoder?
My target AtTiny will be a standalone device (not an in-circuit device), and does not need the Arduino bootloader (which has to manage serial port communications because the Arduino programs in-circuit.
anirbax:
However, I want to save the 1kB of the bootloader, because
my flash memory space is just 2kB.
This was clear from the beginning.
I did forgot about the Arduino being used as a programmer. One of the reasons for that is that I hadn't tried it with different chips and, for someone inexperienced, using AVRStudio to set the fuses is a godsend. Actually seing what each one does is better than writing an hex number that you got from Windows (if other, please correct) calculator.
Depending on the size of the system, having the ICSP pins available for firmware update is quite handy and you should at least consider it. Just because your ATtiny won't be on the Arduino board, doesn't mean you can't have an In-system program. If there's enough space to add the ICSP connector, I'd do it. Trust me on this.