If you're going DAC and amp, the DAC can't function without the amp because of no amplification done to the signal, and in most cases there won't be a TRS output for the headphones. You can get an amp, but you'd have to make sure that the computer and amp have matching inputs and outputs.
If you want virtual surround and the like you should get a sound card. That is mostly important for fps gaming.
If you want the best sound quality for your money you should get a DAC. If you get a DAC an amp is also necessary. An amp for a soundcard might also be a good idea, but not necessarily necessary.
Else a sound card and a DAC does the same, but a sound card has more features and sometimes built in headphone amp. If you don't need those features go for a DAC/amp.
I'd get a DAC-Amp and use that with the HD598 for music, then use an affordable gaming headphone that comes with a USB soundcard for games. It might seem like an additional expense now, but considering earpads on the HD598 are around $50, an extra $100 for gaming headphones now can delay when you spend that $50 for the pads. Not to mention that if it's perpetually on your head the flattened pads might affect the frequency response when you switch to music. Just manually switch which sound device the computer sends the audio through.
I've done this for a while when I got to use my brother's Creative gaming headset, and while the SQ even of explosions sounds better on my audio set-up, I don't notice that while playing since I'm more concerned about shooting the guy shooting at me. Or as I mostly play Total War, directing my spear cavalry to the rear of the enemy infantry and sending the sword cavalry to deal with the enemy cavalry trying to intercept, all while directing bow cavalry to take out the infantry a few at a time as the infantry lines march forward. Basically, with all the stuff going on in the game, the difference in sound quality will not matter.
A sound card traditionally already has a DAC and a headphone amp. The quality may vary of course.
For gaming: If you want proper 3D sound effects and such you might want an Asus / Soundblaster sound card nevertheless. It can be a cheap one or a more expensive one if you want the soundcard to deal with all of that stuff. If you go for DAC/headphone amp, you can either take a digital out (optical or coaxial) from your sound card to the DAC → headphone amp or a combined DAC/HP-amp. That way you'll have the full EAX/whatever effects for the games from the same output.