To connect your Onkyo to the PC through the analog output / input you'll want to use the line out on the motherboard (usually labeled front speaker output or something similar). You'll probably need a 3.5 mm plug to male RCA cable for this.
Sometimes onboard audio is perfectly fine. If you don't hear any noise through the headphone output or line output at the volume you want to listen to, then it's good to go.
Chances are you're going to be underwhelmed with hi-res files. Remastered content tends to sound worse because the trend is to simply make the recording sound louder (Google 'loudness wars'). It's probably best to find a version that's as close to its original release date to get the best sound quality. Oftentimes, that means the material is only available at CD quality.