I've been dabbling with pcsx2 off and on every few months for... a long time now. Unfortunately my computer has remained the same.
Barring any major improvements in the emulator or any of the plugins in the past few months, I can only play with a mild degree of playability. Processor definitely seems to be by far the most important, though I'll admit I'm not nearly as technically-inclined as many here on the Glade. Core 2 Duo 4300, 1.8ghz is not exactly an intimidating processor nowadays. In my testing, some parts of FFX would run smoothly enough, but other parts (especially the transitions between running around and fighting) would slow down a lot.
Still, if my relative dinosaur of a computer can vaguely sort of manage, I would definitely suggest people give this a try. They have a lengthy, fairly detailed compatibility list of PS2 games on their site, if you're interested in any individual games. Though it doesn't list any of the Persona games at all... likely a bad sign.
Emulation hasn't been simple since the SNES era, but this is a pretty sweet program.