For $250, you have a lot of options. The limiting factor for your situation is the sound card though acting as your amplifier. It may make even high impedance headphones play at acceptable volumes, but it won't push the power needed for many of them to reach their potential.
Based on what you said you want (bass), you've got some options.
Sony XB500 (~$50) & Sony XB700 (~$80)
AudioTechnica M50's (~$150)
BeyerDyanimc DT770 Pro 80's (~$200)
Ultrasone Pro 900 (~$300~$400)
I'm biased, but for the money, I'd go with the DT770 Pro's. I probably look/sound like a broken record today with all these Beyer/Bass threads popping up.