When playing some metal on the guitar, I get a lot of background noise. If I slow down the tempo a bit, and really concentrate on hitting the right notes (and only the right notes), the background noise goes away and I can hear what I'm trying to play clearly. However, when I turn up the speed it gets messy (and sloppy).

My guess is that either I'm hitting other strings with my hand, my pickups suck, or probably something else.

I'm using a Jackson SL1 soloist with Duncans for pickups. It's also got a floyd rose.

If this is a skill issue, could someone help me in developing a clear tone? I figure metal is the best way to learn, because if I can do it clearly here I can use it to play other genres like blues, rock, and jazz. Develop good technique now rather than later, right?
I mute with my picking hand, to limit the sound to only the string(s) that I am playing. This prevents the other strings from ringing out and cluttering the sound.
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Yeah, if you're getting it slowly but unclear when fast, just practice it and slowly get it up to speed.

If you've got a SL1, I'm 99% your pups don't suck.
I'm pretty sure this comes down to your playing technique. Slowly build up speed and keep the technique clean. It'll come over time.
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Metronome, make sure you aren't hitting other strings, your pick hand should be moving over a very small area, make sure every note in clear and you press down hard enough but not too hard, and make sure you're rythmn is perfect.
Practice everything slow with a metronome. This is very important if you want to get good.