Should be plenty of guides but for a rough super simplified quick start:
Regular/auto* shot starts the 1 to 5 box thing, value being determined by distance from the enemy during the kill. This starts a chain timer seen underneath the box counter stuff. While the counter is going you can use the lock shot and keep the triggered box value going, regardless of distance, and kills give short extends to the chain timer iirc. When the counter runs out the lock shot just gives 1 boxes.
So the basic way to play is to get a close kill with the regular/auto shot to trigger the 5 box (or highest you're comfortable with getting), then go straight into the lock shot until the timer runs out, then repeat. I think this works on basically everything but end level boss kills. It works on the bosses' non final transformation forms, boss parts, mid bosses, etc.
*Under continuous fire, regular and auto shot are the same, but for a single tap regular shot throws out a longer stream while auto shot gives a short burst. It's useful when you're being more careful about chaining here and there.