You really need a specific USB programmer, which will have a processor on board (which actually does the programming). Using a so called 'serial port' programmer with a USB/RS232 converter may or may not work, such programmers use the handshake lines (not the actual port itself), and latency in the USB to RS232 conversion may cause problems - this is assuming the software will actually work through a converter!.
If you have a proper 'serial port' programmer (with a processor) then a USB/RS232 converter will work fine, and many 'USB' programmers work in this way, being serial port programmers with FT232 converter chips on board.
Here is ICD2 clone USB FT232RL based, uses virtual serial port, not as fast as real USB ICD2, but faster that serial, no power supply. Single sided board.
It's a USB/RS232 converter chip, commonly used to give a simple USB connection - I presume you are aware that the link you posted requires a pre-programmed AVR to work? (as USB programmers are likely to).
The AVR shown on the link is part of the programmer (actually MOST of the programmer), the target AVR is on another board, and programmed while in-circuit.
So you need a programmer in order to build this programmer, a problem with USB programmers!.