Once you've unlocked the Hero Power ability in the main scenario (Legend Log), you can easily build this up. It is technically a level 5 Kizuna gauge (Max is counted as the fourth level, when you can actually toggle Kizuna Rush on). An idea to help you with this, if you want to get through it fast, is to build it up to Level 5 and unlock the Hero Power, then Kizuna Rush with the character, whack some enemies around, and then use
again before it runs out. Usually, it resets the character back to a lower level, but doesn't completely drain them, meaning you can more easily raise the gauge back up as opposed to starting back at Level 1.
(Do note that, should you have made any Bonds with other Crew Members, their gauges will always empty out when you do this, regardless on how you cancel it. If you have, say, Zoro and Sanji in your crew for a mission, and have raised both of them to Rush status, any time you repeatedly use either of them for a Kizuna Rush and cancel it prematurely, while one may have a higher gauge than the other, the second's gauge will always empty out.)