The Phantom Rush requires 3 skill points, which you'll naturally amass from the tutorial and open zone gameplay. Once you've claimed it, you'll be shown what to do in the training simulator.
As you attack enemies, a meter fills up below the combo count. Once it's filled, you've entered Phantom Rush. Continue attacking before the bar runs out and the achievement is yours.
Here's a video of how it works, albeit in the training simulator: (already linked to the timestamp)