Can anyone please suggest how to do barbell overhead press such that kinetic chain stays activated throughout the set? I also need clarification on inhalation and exhalation pattern.
In any exercise performed while standing, the kinetic chain just means the bones, joints and muscles between and external resistance and the floor. If you are standing up, then your kinetic chain is activated. With any part of the chain not activated, you either would not be able to stand, or would not be able to lift the weight. So if you can lift the weight above your head, your kinetic chain is activated. So it isn't really a useful concept. You should probably just ignore anyone that tells you that you need to worry about kinetic chain activation.
As for breathing, holding your breath allows you to brace and keep your torso rigid, which is important when moving the bar. The most common breathing pattern is to breathe when holding the bar at shoulder height, taking a big breath, and then holding it in while raising and lowering the bar. An alternative approach is to breathe at the top of the lift, hold your breathe while lowering the bar, and se a bounce at the bottom of the lift to make the next rep easier.