I typically do a hard flat first serve and then a slice second serve, but I'm curious to know the fundamentals behind adding "kick" to a serve for higher bounce.
The following is a rough, elementary answer. It disregards subtleties such as pronation which are relevant to high level play.
The fundamental mechanic behind a tennis serve is that of throwing a ball. In a basic flat serve, your motion is that of throwing a ball forward. In a kick serve, the motion (for a right-handed player) is of throwing a ball to the right. This is usually enough to force the racquet to hit the ball upwards, generating the desired spin.
I never was able to do it until I heard someone say "Hit the ball up". Once I imagined hitting up on the ball, ever so slightly, I started to get them in consistently, and by the end of the night I hit one over my parter's head!
Once you focus on something so simple as that, all the other pieces fall into place, i.e. brushing the ball, 7 o'clock-to-1 o'clock motion, and where to end your follow-through.
Apparently Naomi Osaka's coach uses an interesting drill to drive this point home: