you need a DC to AC power inverter. you can get these at pep boys, wal-mart, just about anywhere. im not sure what the power requirements are on the PS2 but yu should be able to get away with a 100W one. This just plugs into your 12 cigarette/accessory plug, or if you wish you can wire it up to any other 12v power source. I'd also look for an inverter that has an on/off switch. as far as connecting audio video well thats pretty straightforward. if you have an aftermarket headunit then plug the RCA audio outputs to the auxilliary inputs on your headunit. If you are rolling stock then *shudder* you will have to use a FM-Modulator. Im not particularly a fan of thesedue to SQ, and reliability issues. But go with whatever you need. As far as video, well it appears you dont have a flip-out LCD headunit with a video input, so you will need a LCD monitor, or a small CRT TV if you want to do it that way. If you go with a CRT (regular tube style TV) then you might want to consider a higher powered inverter (say around 300-400 watts). Then connect the S-Video or RCA Video cable to the inputs of the display (CRT or LCD) and you are good to go. Also, consider wireless controllers, or buying extension cables for the controllers and flush mounting the ports into the center console, or something cool like that, so that you can mount the PS2 out of sight.
Hope it all works out for you. BTW took a look at your charger. Dont put underbody neons on it....