You reach elite battle boards by playing Head to Head Blitz Battles. You first choose the state that you want to compete in. After that try to rank up enough points to get promoted to higher boards. The first is Rookie, Second is Veteran, And the third is Elite. I got to Elite after 4 wins. The number of wins may vary depending on how many other players are competing in your same state. But that is not confirmed.
UPDATE: It looks like the promotions are factored from points and not wins. I do apologize I didn't track how many points I had when I got this promotion.