Instructions
Defeat all eight of the gym leaders in the game. Having all the gym badges unlocks the Elite Four challenge.
Beat the Elite Four in the Pokemon League. After you beat the Elite Four for the first time, Jun appears near the Battle Tower area on Saturdays and Sundays for the ultimate rematch. Although his Pokemon will have a level 60 ranking or higher, you can further increase their power levels for a more intense battle.
Trounce the Elite Four challenge 20 times and talk to Jun on a weekend to start a battle. All of his Pokemon will now be between level 79 and 85.