To get 15x, you first need to unlock Veteran by completing Experienced Arcade.
Then, play through Veteran until you unlock the third planet -- this is where you'll be starting new Arcade games during your attempt. If you die or lose your multiplier anytime in phase 3, I'd restart the level. If you die about halfway through the level, you'll still be able to get about 7% by the time you get to the boss.
The boss is a giant square that releases three gray Asteroid-style blocks at three points during the fight. Be sure to shoot every single one of these blocks - focus on the blocks more than the boss. If you hit every one, you'll receive another +5% to your multiplier, bring you up to about 12% by the time you reach level 4. If you don't lose it, you'll get 15% and your pop shortly after entering Phase 2 of level 4.