You can't. The PlayStation 2 will need either the MOD chip installed in order to play copied games, or the Flip Top mod. Flip top mods require you to replace the whole top of the PS2 and boot your PS2 with a bootdisk and then swap the bootdisk for the copied game.
As for as legality goes, well, it's supoused to be illigal even if they are your own copies of games you already own.
I have played on both types of Moded PS2's and I like the MOD chip much better, as you have to do the disk swap, which can get really fast.
As for backing up the games, easy task. Put the game inside your computer, the drive that will read the game, of course. Using just about any program out there, make an image file from the DVD-ROM game to your hard drive, such as *.ISO or many of the other popular formats. Once read and copied, it's as easy as placing in a blank DVD-R to your computers DVD burner and burn that saved burned image onto it from your hard drive. The reason I put the dash format (DVD-R) is that that is what I have used and it works just fine, I've read that the + (DVD+R) doesn't work, I really don't know.
OMG!!!!! It's really that easy.