How does Amagiri Ayato get the nickname Murakumo?
Ayato's nickname "Murakumo" was given by Claudia.

This was revealed in the introduction of Ayato's first Phoenix Tournament match.

From the light novel The Asterisk War, Vol. 3, Ch. 2 - The Phoenix,


  “The first two onstage are Ayato Amagiri, Seidoukan Academy’s number one,
  and Julis-Alexia von Riessfeld, number five! Now, Amagiri only rose to the top a
  few weeks ago in a duel against the former top-ranked student! He is a truly
  brand-new star—so new, in fact, we still don’t have much data on him
  ourselves! Ah, although we did hear his nickname Murakumo came from
  Seidoukan’s student council president, Miss Enfield, herself.”
  
  (Emphasis added)


However, the reason for the nickname itself was not stated anywhere.