How do I make most use of the Adrenaline Shield?

Overall the Adrenaline Shield is more useful as better you are and/or the situation is for killing enemies successively. You're usually not able to be under fire and able to maintain your shield by killing enemies because of the low TTK (time to kill) of Planetside 2.

It requires a more aggressive mindeset and if you play defensivly Adrenaline Shield is not the right tool for you. For the average player the use of Adrenaline Shield is limited to situations where you are able to kill enemies in successive without getting to much incoming fire.

Examples the Adrenaline Shield may be useful:

Attacking/Defending a Tower Base (using a shotgun) on the 2nd or higher floor.

Roaming a Tower Base may give quite some easy kills if you keep mobile. This allows you to keep your shield up and stay mobile.

Defending a position with team mates.

With good positioning you may be able to avoid incoming fire and kill enemies so you may be able to keep your shield high and survive overall longer.

In terms of mechanics, the only differences between Nanite Mesh Generator (NMG) and Adrenaline Shield are the passive charge rate and the instant charge from kills. A maximum rank NMG will fully recharge from empty in 45 seconds, while all ranks of Adrenaline Shield require 60 seconds to recharge from empty. This means that if you gain 20% charge on kill while using the adrenaline shield, you gain the equivalent energy to 12 seconds of charge instantly. If, when engaging an enemy and surviving, you also get a kill (by far the most common outcome) Adrenaline shield will leave you with more energy than you would have from NMG for the next 44 seconds, and both are equal at 48 seconds. Lower ranks decrease the time window in which you come out ahead with either shield.

So to answer your question: the best way to make use of the adrenaline shield is to make sure that you get a kill when you engage the enemy. Avoid fights where the enemy can choose to disengage after you trade shots and lose your shield energy. In general, this means engage in close combat scenarios or when the enemy cannot get into cover quickly. Something that may improve your performance in close quarters scenarios is weapon selection: I recommend shotguns, SMGs, or the close range LMGs (Orion, Anchor, MSW-R) with laser sights.

Understand that even the most skilled players cannot kill fast enough to keep the Adrenaline Shield charged, but they can get the extra charge faster, which saves them from one bullet and may be enough to change the outcome of their next fight. The only situations that can lead to a net gain in shield energy are picking off unaware targets with good positioning and holding down a choke point that the enemy run single file into your fire.

Falling back to let your NMG or Adrenaline Shield recharge is almost always a poor choice. While it's true that you are less effective fighting without full charge, with good spawn points you can often get back in the fight after hurting an enemy and dying faster than waiting for your shield to recharge fully, even with a maxed NMG. While it may not improve your stats, it will be more effective overall in the fight and of course be more interesting than hiding in a corner for a while. And who knows, you might even win the engagement without full energy - after all, you have the same normal health and shield as any other class.