How is the most natural sounding way to phrase this question, which is a little odd in a way because you could literally answer "How would you rate yourself on the scale from 1 to 10" with something like "I would pick a number between 1 and 10 that seemed to match my self-image in this respect" and be technically correct. You probably wouldn't get the job in most cases. That ambiguity doesn't exist if you phrase the question with "What" but even so a native English speaker would choose the former every time.
What would fit into the middle of a sentence better:
"On a scale from 1 to 10, what rating
would you give yourself?"
I would tend to use a scale from 1 to 10. The only situation where I can think I would use a scale of would be to describe the size of a map or diagram- "This map is drawn at a scale of 1:20000" - if I'm describing a range it would always be "from 5 to 100."
Edit: Also notice that it is always a scale from 1 to 10 rather than the scale. You would only use "the" if you were describing a particular scale - for example "What is the wind speed on the Beaufort Scale?" In this case we are using an abstract scale that is defined by the question, so it is "a scale".