Spend most or all of your budget on the headphones. With your budget you can get a very high-end one from the top brands. If you stick to 300Ohms or less, any headphone should be drivable from the onboard soundcard. On-board audio is amazing these days, so unless you get noise or low output from that, don't spend any money on DACs and amps.
You could spend some money on a (USB) soundcard with Dolby Headphone support (Asus U7 for example), that will get you good surround sound from a stereo headphone. Don't believe that's possible? Listen to the virtual barbershop with headphones.
Don't bother with hi-res audio (you're not a bat and you can't hear a mosquito next to a jackhammer) and lossless audio (except for archiving purposes), it's mastering that makes a lot of modern music sound atrocious. Find well mastered music, enjoy it on your high-end headphones fed through a $2 sound chip.