I have recently been looking into increasing the speed of my playing; however, I look at guitarists like Synyster Gates and Paul Gilbert and am baffled by how quickly and cleanly they can play.

I was attempting to learn Technical Difficulties and The solo for Beast and the Harlot and ran into problems.

Following the advice of many others, I begin practicing the songs slowly, and attempted to gradually increase speed. The problem is that I hit a barrier in which I can no longer get any faster. I've been playing for about 3 years, yet I've failed to see a dramatic increase in my speed for a while.

I seem to have a lot of trouble with my alternate picking, yet no matter how slow I start and how gradually I increase the speed, I fail to see improvements. Can anyone give me some tips on how I can combat my problems, how to get past the speed barrier?

It's probably a technical issue. Look at those guitarists whose speed impresses you. Their picking hand motions are very small and economical. That kind of minimized motion is one of the key elements of their playing. If you're not addressing the size of your pick strokes, you can practice for another 3 years and see little to no improvement.

When you practice, watch your pick strokes and try to make them as small as possible. Keep practicing that smaller motion until it becomes muscle memory to make that smaller motion. Then start speeding up, getting yourself acclimated to using small pick strokes at higher tempos.
If you don't already, use a metronome. It is the key to speed and timing.

When you increase speed it should be in small increments and you should NEVER be sloppy. if you begin to play sloppily, you're going too fast.

Be patient with yourself and up tempos slowly. I know you want to rush and get right up to full speed but it's all about muscle memory and doing things right, which takes time.

As Geldin mentioned, minimize motion. economizing your movements will make a great difference in your playing. Aim to be as efficient as possible.