Seconding what Schlegel says here. "Just enough to graze over the string to down pick the note & then back for the next up stroke."

When picking fast for this sort of thing you probably won't be digging way into the string -- it's just the very tip of the pick. It's a tiny arcing motion rather than a straight up-and-down. (Exception here might be Paul Gilbert as he seems to be able to pick really hard AND fast but he's a freak.)

Other area to think about is which muscle groups / joints you are using.

Similar to when walking / running, there's going to be a shift between different major muscles depending on the speed of picking. For instance, when you walk it's a different set of muscles and motions than when you're running. They overlap, and it's the same with picking. At a very slow tempo it can totally be wrist.

When you reach a significant speed there's probably going to be more bicep involved. If you try to use only your wrist it will probably get exhausted pretty quickly. Michael Angelo Batio locks his wrist firmly and goes entirely bicep at high speeds. (Exception here might be Yngwie but he's Yngwie.)

But yeah, the key is just take a metronome, set it at (whatever slow tempo) and play cleanly. Increase it by 1 or 5 bpm every few minutes. When it gets to be too hard, slow it back down, then increase again. After many days / weeks / months you'll see the progress! This is how I have increased my dexterity time and again.

-Carl.