I can't take credit for this but for the sake of expediency, here you go.
The Botany Bay is found by Starfleet in the vast reaches of space.
Admiral Marcus, the head of Starfleet, awakens Khan to have him assist Section 31 in the secret creation of ships and weapons, preparing for whichever enemy the Federation may face next. The John Harrison alias is created to conceal Khan’s identity.
Harrison bides his time, working for Marcus. He designs new long range torpedoes, inside of which he smuggles his crew; hoping to be able to escape and then begin taking over the galaxy. Harrison may work with Marcus but he always has his own agenda throughout the film.
Admiral Marcus finds out that Harrison has hidden his crew in the torpedoes (This is where Harrison begins to go off the rails for Marcus, until then everything has been according to their plan. Harrison begins enact his own plans at this point while still following most of what Marcus and he had laid out.)
Meanwhile, the Enterprise crew is on a surveying mission. A volcano is going to destroy the planet and, in the process, a primitive society. Spock has created a cold fusion device that, when detonated, will render the volcano inert. As Spock lowers into the volcano, Kirk has stolen an item the primitives worship in order to lead them out of the volcano’s kill zone. He and Bones escape by jumping off a cliff into the sea where the Enterprise is hidden. Spock has been abandoned by Sulu and Uhura because the shuttlecraft they are in has been disabled by the volcano’s ash. Spock sets the device to go off and Kirk rescues Spock, an action that violates the Prime Directive since the natives see the Enterprise rise out of the ocean on its way to save Spock.
Marcus wants a war with the Klingons, so he has Harrison attack London. Harrison convinces a desperate Section 31 officer, whose daughter is dying of an incurable disease, to bomb a secret Section 31 base in London. In return, Harrison gives the officer a cure for the dying girl. Harrison has himself identified as the main villain.
Harrison allows himself to be photographed leaving the scene of the crime.
Kirk has the Enterprise taken away from him, because of his hubris. Pike is given back the Enterprise and fights to have Kirk made his first officer. (Marcus convened the meeting that Pike could not be at, this is part of his plan.)
The bombing puts Starfleet on high alert and protocol requires that all senior staff, captains and first officers of Starfleet’s closest ships, meet at Starfleet HQ.
Harrison attacks Starfleet HQ because this leaves Marcus fully in charge of Starfleet (Khan’s ship was scanning people before shooting them so only the people Marcus wanted dead would be killed, like Pike, who would stand against Marcus in his warmongering. This is also the reason Kirk was busted down to first officer, Harrison was scanning and killing captains and other admirals. Those left would be the one’s Marcus though he could control).
Harrison beams to Kronos. Harrison would seem to be in league with the Klingons and Marcus would get his war. Harrison is using Marcus and Marcus is using Harrison. For Harrison this gets his people away from Marcus and gives him a chance to save them.
Kirk, Scotty and Spock find that Harrison is not on Earth anymore but on Kronos because the are suppose too.
Admiral Marcus says that he does not want Kirk to go capture Harrison; instead, he gives Kirk 72 torpedoes that only Marcus knows contains Harrison’s crew. Once these are shot at Harrison from the Neutral Zone, two of Marcus’ problems will be eliminated in one fell swoop. Kirk is given back the Enterprise (thinking Kirk will be someone he can control easily now that Pike is dead. When Marcus says that Pike’s blood is on his hands he means literally). Kirk is a pawn in Marcus’ plan to start a war with the Klingons. Kirk will fire the torpedoes at the Klingons, the Klingons would go looking for the perpetrator. Unbeknownst to the Enterprise crew, Admiral Marcus has the warp core sabotaged. The crippled Enterprise will have no choice but to fight back and the Klingon empire will be outraged. War will begin all because of Kirk, a renegade captain on a mission of vengeance. This is the reason that Marcus, when Kirk and Spock visit his office dismisses everyone else. This does not come from all of Starfleet, just Marcus. (These are all things that Harrison is well aware of; it is in the novelization)
Carol Marcus forges orders to get herself transferred to the Enterprise. She is a weapons expert and the daughter of the Admiral. She knows he is up to no good since all records of the torpedoes disappeared after she confronted her father about them and now they have resurfaced on the Enterprise.
Kirk, listening to the advice of Spock and Scotty (especially after Scotty leaves the Enterprise to protest the torpedoes that he is not allowed to inspect and therefore finds too dangerous to use) travels to Kronos to capture Harrison, instead of using the torpedoes to destroy him without standing trial.
Kirk has Sulu threaten Harrison with obliteration by the torpedoes if he does not come quietly with the landing party.
Harrison, hearing Sulu’s message knows he has been right about Kirk and Spock, they will not use the torpedoes. He then asks how many torpedoes Kirk has. At hearing the correct number and knowing they are the same torpedoes that harbor his crew, he surrenders to Kirk, and bides his time till he can find a way to save them. He is playing Kirk the whole time.
Kirk, Spock and Uhura return with Harrison to the Enterprise. Kirk tells Uhura to inform Starfleet that they have Harrison in custody and that they will return him to earth when the warp drive is fixed.
Harrison, in the Enterprise brig, tells Kirk to open the torpedo so he, Kirk can see why he, Harrison did what he did. He gives Kirk a set of coordinates and tells him there is something there that will verify that he is telling the truth.
Kirk contacts Scotty, who is on Earth, and asks him to search the coordinates Harrison has given him to see what is there.
Kirk has Carol Marcus and Bones open the torpedo. They find a cryogenic tube harboring a 300 year old man.
Kirk quizzes Harrison and finds out that he is Khan from a time long past who has been used by Admiral Marcus to create weapons for a coming war with the Klingons. Khan is also aware that the Enterprise has been sabotaged by Marcus. Khan knows what Marcus is planning: everyone’s death and a war with the Klingons. Again, this story is only half truths, Khan it playing Kirk to earn his trust.
Marcus shows up in the Vengeance. He tells Kirk that he made a mistake in waking up Khan and that he is only trying to rectify his mistake. He asks Kirk to give him Khan. Marcus even tells Kirk the truth that Khan is playing him.
Kirk, no longer trusting Marcus, warps to Earth in an attempt to escape and return Khan to face judgment. Kirk thinks that he will be safe at warp because Marcus will not be able to catch up. Carol Marcus is reminded by Khan that Kirk is not actually safe at warp because the Vengeance has a new, faster warp drive. Admiral Marcus will catch up and destroy the Enterprise.
Carol believes that Admiral Marcus will not destroy the Enterprise with her aboard, so she has Kirk allow her to contact her father.
At this point, both ships are near Earth’s moon because the Vengeance has knocked the Enterprise out of warp and heavily damaged it.
Admiral Marcus proves to be the villain and transports Carol to the Vengeance so that he can finish off the Enterprise. He plans to tell Starfleet that Kirk went rogue and joined with Khan and he had no choice but to end them both by destroying the Enterprise.
As the Vengeance powers up its weapons to level the final blow, it loses power.
Scotty contacts Kirk and tells him that he has stowed away on the Vengeance and disabled the ship for 3 minutes.
Kirk, knowing that they cannot destroy the Vengeance from the outside, asks Khan to space jump with him to the Vengeance so they can take it out from the inside (Kirk can only take minimal people and needs Khan knowledge of the ship and his physical prowess). The Vengeance has been created to be run by minimal crew and one person if needed. Khan now has the opening he has been planning for.
Scotty, Kirk, and Khan take over the Vengeance. Kirk, knowing he is being played by Khan, has instructed Scotty to stun Khan the moment they have taken over the bridge. Scotty does so but Khan, being genetically engineered has faked being stunned and very quickly and disarms Scotty, beats up Kirk, breaks Carol’s leg, and kills Admiral Marcus with his bare hands.
Khan contacts Spock and tells him to give him his crew, whom he still thinks are in the torpedoes, in exchange for Kirk, Scotty, and Carol.
Spock agrees to this. Unbeknownst to Khan, Spock has removed Khan’s crew from the torpedoes and activated them (the torpedoes) so that they’ll blow up in the Vengeance’s cargo bay (Khan underestimated both Kirk and Spock in his plans. Superior minds breed superior arrogance). This disables the weapons and major systems of the ship, but not before it’s knocked out the Enterprise’s warp core.
Powerless, the Enterprise tumbles towards Earth because it is close enough to be caught in its gravity.
Kirk and Scotty, who Khan has returned to the Enterprise, race towards engineering so (that) they can restart the warp core.
When they get to engineering, they find that the warp core is out of alignment and the only way to fix it is for someone to readjust it from the inside, sentencing that person to death from radiation poisoning.
Kirk knocks out Scotty and manages to save the Enterprise, but dies as Spock watches beyond the protective door.
Khan use the last bit of power in the Vengeance to ram the ship into Starfleet HQ, located in San Francisco.
Spock, realizing that Khan could survive the crash, has Sulu scan for lifesigns on the Vengeance and is then beamed down to Khan’s location.
A mad chase through San Francisco ensues till they end up on top of a transport where Spock tries to subdue Khan.
McCoy has Kirk in sickbay. As the crew visits his lifeless body, a dead Tribble that McCoy had earlier injected with Khan’s blood comes back to life. McCoy, realizing that Kirk can be saved, transfers Kirk to a cryotube to save his brain function. McCoy informs the bridge that Kirk can be saved if Khan is captured.
Uhura beams down to the transport where Spock is being beaten by Khan. She repeatedly stuns Khan which gives Spock enough time to tear off a piece of the transport and finally knock Khan out.
Kirk is saved and a year later the Enterprise is rechristened as it heads out on the first ever five year mission.
