The most non-invasive means of running homebrew on a ps2 is the SwapMagic / Slide Card combo. There are also no-solder modchips which allow you to swap disks without a slide card. Just google the terms I've mentioned and you should find everything pretty easily.
Also, be aware that I am writing this with the older ps2 models in mind. I am not sure what is necessary to perform a swap in a newer ps2. I'm pretty sure that Dcemu has a ps2 homebrew section. You should search for a guide in that section's forums.