This is a story trophy and cannot be missed!
The best way I found to defeat the pair was to focus entirely on Elektro until Vulture would try to rush you, when the latter does that do what the game asks you and then move back to Elektro. Reason being that Elektro is gonna be a constant pain in the a** unless you take care of him first and also the fact that Vulture is a much easier boss to beat when you aren't being blasted by lightning every few seconds.
Having some gadget skills unlocked helps with beating Elektro, especially the one that slows down time when you aim.
Just overcharge his batteries and he'll be down in a quick manner leaving the nearly helpless Vulture to become the prey.