I want to start running this game and I'm wondering how I can play KH-J on a American PS2.

Its kind of a pain to try to run JP games on American PS2s, i recommend if you want to play J to just invest in either of the actually native Japanese consoles. If you can't or don't want to, you'll either have to install a modchip like ModBo or Matrix that removes region locking, or you'll have to use FreeMcBoot to run a Swapmagic ELF and utilize that to play the Japanese version.

Ok thanks.

I personally believe running Japanese games on North American consoles to be very easy using the mechapwn exploit. All you will require is a Free McBoot Memory Card and a USB stick. There are steps online on how to do this (link below). For anyone else who may reads this PAL consoles also work for this exploit but has to be a slim console, 50k or backwards will not work for PAL. However if you have a Japanese or North American unit that's a 50k the exploit works. It essentially turns your console into a debug console which allows all regions to be read.

https://github.com/MechaResearch/MechaPwn

You can now super easily make any PS2 from SCPH-5000X - SCPH 9000X model into a region free console with Mechapwn. Use this build of mechapwn specifically to avoid issues with game compatibility. https://github.com/MechaResearch/MechaPwn/releases/tag/3.00-rc4

Tutorial for using MechaPwn: