IMO you will be much happier splitting the 650$ over a good amp and a good dac rather than spending it all on an amp and using your computer's line out. For me, it is easy to tell the difference between the built-in dac of my Sony Vaio Z and my iPod Touch 2G, MF VDac and Metrum Octave. It goes something like this:
($free) Vaio Z built-in sound: ****ty, grainy... just terrible.
($free) iPod touch 2g: decent, fine for on the go.
($200) VDac: decently good, weakness is the highs
($1000) Octave: awesome
This is through a CK2-III that I built, which is a pretty good DIY amp that probably would cost around 600$ retail. My usual headphone rig is the VDAC+CK2-III (the octave is really for my full system). And yes, I run HD650's.
Another question though is what kind of music do you listen to?
A good strategy that I wish I had followed when starting out in the high end audio world is to buy used stuff - only popular models so it's easy to flip it if you don't like it. This allows you to discover what you like without breaking your bank account. For example, I went through 3 or 4 iems before settling on the IE8's.
Edited by Zynec - 2/20/13 at 10:21am