MAN OF STEEL
(Hopes/ what I would have)
Krypton was not destroyed naturally but invaded by Braniac and destroyed.
MAN OF STEEL 2
Braniac ( or Darkseid) discovers the remaining Kryptonian along with Earth and the Man of Steel faces off against the destroyer of his race to defend his new home.
Braniac uses a Kryptonian creature that he had captured and experimented on (maybe someone meaningful to Kal El) and creates the ultimate weapon: Doomsday, who faces off against Superman.
Doomsday kills Lex Luthor's fiancé who Lex Luthor (as a friend of Superman), told Clark/ Superman to protect. But Superman is caught off guard by this since he was busy stopping Braniac (who falls before his unstoppable creation) and saving Lois and/or Billy. Lex Luthor is furious after this and decides that Superman does not deserve his power.
MAN OF STEEL 3
Lex Luthor (who has been Superman's friend throughout the movies ends up pitted against Superman when he realizes that the Earth must be conquered to be protected, that he should be the ruler, and that Superman (who he knows is Clark: his friend) can not be depended on, since he fails to rescue Lex Luthor's fiancé (who dies at the hands of Doomsday). He launches a political campaign, becoming president of the United States. Then he secretly plans to conquer the world and kill Superman (his hatred and jealousy has grown immensely at this point) He convinces the public to be angry at Superman since though he saved the world, he was not able to stop the huge losses of Braniac's invasion.
Superman must now stop his former friend from his evil goal (details/plan for that is in progress)
He builds a high tech battle suit with a kryptonite core (can shoot kryptonite, which is a laser not a crystal and does not kill superman but hurts him on impact (it is also not specifically deadly to Superman but was specially made to be capable of hurting him after Lex studies his blood from the doomsday battle.) he names it kryptonite to hit a personal note.
Soon Lex makes Superman public enemy number one, making him look dangerous to the public.
Lois Lane and Superman and scheduled to be married (or are married) in the movie, thus making Lex hate Clark more for the loss of his own fiancé.
Keep in mind this is just a quick slab of ideas, I'll update this later.