To play/alternate pick fast and accurately, you need to slow things down until you can play and repeat the passage flawlessly. Don't practice mistakes or sloppy notes! Speed will come naturally with time.

Another important thing is that you play slow exactly like you play fast, or else once you start speeding up, the picking motions might change and you'll hit a speed barrier that you can't seem to pass.

To make sure, just alternate pick a single note as fast as you can and start slowing down, without changing the picking motion.. then practice your passages with THOSE picking motions.