I keep seeing everyone asking about how Klaus died in the first Apocalypse, if Five buried him alive, etc, etc. While I see a lot of theories and thoughts out there I have yet to personally see anything similar to my own take on this plothole and seeming lapse in judgment from the writers.
I do not think Klaus came back to life after the initial Apocalypse, which will simply be referred to as "The Apocalypse" henceforth, and I especially don't think Five buried Klaus alive and I will briefly explain why.
First and foremost there is no way that Five wouldn't have noticed. He was thirteen when he discovered the bodies of his siblings in the wreckage of the world and I'm sure we can all come to the conclusion that Five was arrogant enough to think he'd get back to his time far quicker than he actually did. He would have lingered around their bodies indefinitely for at least a small amount of time before he started to diminish the resources in the area and there is absolutely no way in hell that if he did bury them that he would have dug a deep enough grave or buried his siblings deep enough that Klaus wouldn't have been able to either dig his way out or make enough noise that Five would take notice, regardless of how many resurrections it would have taken to eventually get Five's attention.
However, despite all that, I don't believe Klaus came back at all after his initial death in his futile fight to stop the Apocalypse.
From all the deaths we've seen or heard of when it comes to Klaus he was either completely unaware that he'd died at all or he was already aware that he had the ability to resurrect himself. This does include his stint in the afterlife when he died during the rave in season one, as the only reason he believes he died is that he met God and talked to Reginald and that could easily be brushed off later as a side effect of his powers during a "near" death experience. For all intents and purposes, Klaus would have no reason to think he's going to stay dead in any scenario, especially given just how doped up he'd been a majority of his life. Subconscious or not, there is an effort put in to come back from the dead because Klaus sincerely believes that he's just going to bounce back or "wake up" as if nothing had happened at all. We have yet to see a Hargreeves with abilities they don't activate themselves unless we want to assume that Luther and Marcus are just in a constant heightened state but even then their powers really fall under a different category when compared to everyone else's. I firmly believe that Klaus' ability requires an effort because we've seen him remain in the afterlife or "the Void" after he kicked it in Oblivion. He came back when he more or less wanted to after Luther's relentless pestering, not because he was just sucked back into the realm of the living.
Long story short, I don't think Klaus came back after the Apocalypse because there is no way that man watched the world explode and thought "Oh, I'll walk away from this." Klaus really thought he died, and he had zero reason to believe otherwise because Reginald never thought to inform him about the extent of his abilities, and so, by all means, he really did.