I cover all that in this tutorial on building speed with a focus on tremolo picking.
https://www.guitartricks.com/tutorial/287/
You've got the right idea with grazing the string. Going too deep into the string inhibits speed. Essentially, use minimal motion & stay relaxed. Wide motions & tension inhibit speed. Hope that helps!
Christopher Schlegel
Guitar Tricks Instructor
Christopher Schlegel Lesson Directory