Twist 1: Cortana is somehow alive (which makes little to no sense, given the debris we see strewn around Chief at the end of Halo 4 - if "the Didact's ship" was "pulled into slipspace" then what was all of that debris? What did that nuke ACTUALLY blow up?).
So yes -- people were expecting her to either stay dead, or to have a legitimate reason for being alive. There were several avenues by which the character could have returned without any hand-waving. The only thing that makes sense to me about that (without requiring something akin to a storyteller's retcon) is that this is not the Cortana from the Didact's ship, but rather one that comes from the Chief's interactions with The Librarian in Halo 4 (remember when you first lost her at that data terminal? Immediately prior to your conversation with The Librarian's imprint? Maybe The Librarian's imprint made a copy or something. Or maybe the Cortana that died WAS the copy. *shrugs*).

Twist 2: Cortana is not feeling like herself. In fact, she's feeling very Didact-y. So much so that she's going to impose her version of peace on the galaxy under penalty of death, and she's not willing to discuss it any further.
I am suddenly reminded of Cortana's Halo 4 dialogue where she says
"It won't be me.
...you know that, right?"

Twist 3 (and the False Advertising): Chief and Locke aren't actually diametrically opposed to each other. The marketing -- heck, the cover art of the game AND the intro screen that you see every time you boot up the game -- all show Chief and Locke facing off against each other, as though they've taken two opposite stances on some philosophical issue that they both believe that they're fully informed about. All of these elements seem to paint the entire story of Halo 5 as revolving around this "central conflict" between these two characters, and it's all bogus. Locke doesn't even dislike the Chief, he's just doing a job.

Twist 4 (also False Advertising): We never see poncho Chief on The Ark. The Ark never even comes into play at all. Nor does the poncho. I'd like to remind everyone that they went so far as to make an action figure out of poncho chief. Poncho Chief was a thing.

Alright, here's a barebones articulation of what you're asking for:

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Alright, look: in Halo, there are these beings called Precursors. They're supposedly the first beings to have ever lived -- as in: they're literally the oldest living creatures. They are billions and billions (and billions) of years old. They're mind-bogglingly intelligent and powerful, and have successfully kept most details about themselves secret.

They created this thing called The Mantle of Responsibility For All Life, which is basically a position of guardianship (kinda like guardianship in a custody battle over a kid) over all life. Now, what's important to remember is that only The Precursors get to say who inherits The Mantle from them and who doesn't. If you don't like that, tough, cuz they don't care, and you can't stop them. No, like, seriously, you can't. The most powerful race in the galaxy tried to back at the height of their power, and they got smacked down so hard that there hasn't been a smack down before or since that has come even close to comparing with it. It was the undisputed mother of all smackdowns.

They also created two races of beings in their own image, with the intent of these races inheriting The Mantle: the race that we call Forerunners, and the race that we call Humans.

The Forerunners were made first, and as they got older, the Precursors became dissatisfied with them, so they told the Forerunners "Dudes, we changed our minds: after watching how y'all got all lame and stuff, we've decided that we're gonna hand down The Mantle to your new baby brother, Humanity."

The Forerunners at that time freaked the frick out. So much so that they carried out a holocaust against the Precursors' physical forms, and thought that they had pretty much killed them all. (Pretty sure that you can't actually KILL a Precursor in the most absolute sense, but that's a whole 'nother can o' worms that'll have to wait 'til another time, 'cuz it gets pretty complicated pretty quick.) The Forerunners who survived carrying out this holocaust left the galaxy in horror, and never came back. The Forerunners who stayed home did a massive cover-up about the whole thing, and erased all traces of it from history.

Now, fast-forward 10 million years. The Forerunners that are alive at this time don't know jack-diddly about any of that Precursor stuff, same as you don't know jack-diddly about what was going on in this galaxy 10 million years ago. They don't know what The Mantle actually is, they only think they know (they ballpark it), and so their version has serious problems. It comes to pass that for various reasons, they go to war with Old Humanity and decide to de-evolve us instead of making us go extinct.

These Forerunners encounter The Flood. It forces them to activate these Forerunner-built weapons called Halos as a last-resort to prevent The Flood from succeeding in killing absolutely everything.

The Halos activate, and kill ALMOST everything (which is an infinite difference from ACTUALLY everything). For various reasons, the Forerunners were not able to make it to The Ark (a Forerunner installation outside of The Milky Way) before the Halos fired, and so they all died (with a few exceptions). Afterwards, what little sentient life is left is sent out from The Ark to locations throughout the galaxy to start re-populating the place. One of the races that was sent out was the de-evolved version of Humanity. Some of the other races that were sent out end up meeting up and forming The Covenant.

Fast-forward 100,000 years.

The Covenant scans a planet and discovers Humanity. A grand total of like, three members of The Covenant find out that Humanity is supposed to reclaim the Forerunners' legacy (we are ID'd as "Reclaimers"), as well as all of the Forerunner artifacts that the Covenant races have worshipped for so long, and so their response to finding this out is to freak the frick out and pursue our total annihilation (sounds familiar).

These ~3 Covie dudes don't tell anyone else this information about humanity, they just all keep it to themselves, and so whenever members of The Covenant would ask things like "Hey: So why are we just straight slaughterin' these 'Human' dudes again?" the ~3 Covie dudes would all just be like "Look, dem dudes is straight unclean and such, and so we've marked 'em for DEATH, and... that's... really all you need to know. Aight? Now get to glassin'."

So they waged war against us for ~27 years.

Then The Flood showed up again, making things complicated for The Covenant, and because The Flood showed up when they did, things worked out for us so that we survived the war against The Covenant. Which is a funny little thing, because if the Flood hadn't shown up and been such a problem for the Forerunners, we wouldn't have survived their war against us either.

Fast forward to Halo 5.

After the nuke goes off at the end of Halo 4, Cortana (or something like her) has it in her head that AIs (The Created) need to be the ones to claim The Mantle -- not humanity or anyone else. This ain't gonna work though, because only The Precursors call that shot. She doesn't know this, though, so she's gonna go through with her attempt anyway. For some reason, she calls out to Chief to see if he'll support her, and when he shows up, she locks him safely in a cryptum so that he can't interfere with her plans. Fireteam Osiris shows up with a Forerunner Monitor at the last minute and snatches the Cryptum outta Cortana's grasp before she leaves to go lead the war effort in pursuing this whole Mantle deal.

As for Halo 4, The Didact (specifically, the original Didact, called Ur-Didact) was put in a cryptum by The Librarian 100,000 years ago, pretty much right before the Halos fired. [The Didact was the military leader of the Forerunners, and not just against the Flood, but against ancient humanity as well. He lost his kids in that war, and that source of resentment is added onto the fact that he already strongly disliked humanity.] The idea was that he'd be able to do some sort of a "communing with The Domain" sort of thing from within the cryptum for a good long while, and that doing so would "straighten him out" so that when he was finally woken up by humanity at some point in the distant future, he'd be more like a friendly advisor and less like a narrow-minded murderer. As you can see, it didn't quite work out like that. Main reason for that is that he wasn't able to commune with The Domain during those 100,000 years, and so you can understand how he might be upset. Imagine being locked in a escape-pod with no internet or anything for 100,000 years. Frankly, I think I'd want to kill everybody too, if I'm being perfectly honest with myself. [We don't know what The Domain is, exactly, but current theory is that it's actually the domain of the Precursors, who store up all kinds of knowledge and memories and other such things (such as minds). The Forerunners didn't fully understand what it was, exactly, either, but they used it kind of like a Forerunner version of The Internet. It actually kinda seems like a good/happy version of The Gravemind, but way, way bigger.]

So yeah: the "tl;dr" version is:
The Precursors are trying to go through the process of handing down The Mantle of Responsibility for All Life to humanity, their chosen Inheritors, and stuff keeps happening to get in the way of that. Whoever has gotten in the way of that happening has gotten themselves thoroughly Dealt With ™.
Current theory on Cortana's behavior is that Precursors are causing her to behave the way that she is so that they can test humanity's eligibility for The Mantle. It was never the Precursors' plan that she and the other AIs actually succeed in claiming The Mantle, they just want to test us in some specific way. For more on that, check this out: http://carnage.bungie.org/haloforum/halo.forum.pl?read=1209604

Hope this helps. Whatever questions you've got, I'm more than happy to answer.