The reason the discs jump in some players and not others is down to the quality of the laser in the player and also the way it's read. If you copy the files from the disc onto the PC it won't jump as it can read it over and over again to try and read through the scratch and if it can't read it, you'll get a file copy error after quite a while. If it's just played normally, it doesn't have time to read it all these times so it will jump. So the best thing to do is to try and rip the DVD with a program like DVD Shrink (click here) and see if this will let you rip it without any errors.
If there are still read errors, then all you need is a capture card, a composite video & audio cable (the one with 3 wires - a yellow, a red and white one) and the software will come with the card.