I wouldn't worry about upgrading the DAC if I were you. If you play songs that are in 44.1/16, you won't see any benefit of upgrading the DAC. Probably the best, most trouble-free method of upgrading your rig is to get a FiiO L9. It's an LOD cable for any Apple devices pre-IP5 (unless you have a lightning adapter), and it will allow you to achieve a pseudo-line-out from the device. It will be louder than the headphone out, but you will still be able to use various eq effects, which is a sign that the amp is still being used (hence, the word pseudo-line-out).
There's no way to achieve a true line-out on the iPod Classic. Well, I don't think there is. I remember seeing an iPod Classic with line out cable sticking out behind the player (they soldered the cable somewhere on the PCB to bypass the amp, I think), but that's just too much work, imo. You can also take a look at
RockBoxing your iPod Classic. I'm not going to go through the reasons why because the link I gave you will describe everything for you. Personally, I think that it should be a requirement for anyone who has a Classic.