How can I copy my PS2 saves to my PC?

This can be done with the uLaunchELF browser. uLaunchELF is homebrew software, so must be run on a modded PS2 or via other methods for running unsigned software (the same techniques can be used to run pirated games, so I won't go into detail).

Once in the browser, copy your saves from mc0:/ or mc1:/ (the first and second memory card slots) and "psuPaste" them to mass:/. This will produce files which can be used with mymc to produce PCSX2-compatible memory card images.

If you have access to a PS3 and a PS2-memcard-to-USB converter, you can copy the PS2 saves to the PS3. Then, from the PS3 you can copy the saves to a USB drive which should be accessible from a PC.

I can't say if the resulting files are compatible with PCSX2 though.

Second @EmSixTeen's answer re: uLaunchELF. You say you don't want to mod ... I don't think you'll be able to do this without some kind of modding. You might look into FreeMCB and OPL, they are handy 'softmodding' tools that install a mod on your memory card - so, no dealing with mod chips, at least. You'll probably need something like a GameShark / CodeBreaker to install it though - and I believe you can just use that (GameShark) to copy your saves.

(If you decide not to box up the PS2, the homebrew stuff is pretty handy - yes, even for non-piratey uses.)