There are a few parts to your question, I'll try to address them all (as well as I understand your brief plot synopsis), but not necessarily in order.
First, my unpopular opinion:
Frame Challenge: Prologs are boring and over-rated
The boy starts the book off by describing what his world is like and
the rules and power it is constructed on...
This sounds like world-building. Write it down for yourself, but don't let it take over the storytelling.
Stories are made of characters and their immediate conflicts..., not histories of socio-economics, or textbook lessons about people who are not in this book.
The worldbuilding iceberg is 90% underwater, meaning the part that sticks up that everyone sees implies a lot more 'world' that we don't see. You will have many opportunities to show-don't-tell about your world since your protagonist is about to get a first-hand look at how the sausage is made.
There is another problem that prevents using a prolog here. Prologs are 'factual' because they come straight from the greatest authority: the Author. If you straight up tell the reader what happened, they have to accept it.
However, if you leave the history as clues the reader can piece it together and fill in the rest. With a prolog history is hard-set, without a prolog history is open to interpretation.
Protagonist's POV is biased
If Assassin Boy described his world at the beginning of his own journey, he would be all wrong anyway.
He is coming from the indoctrinated environment of his childhood – where morality is black-and-white, and he exists for 1 purpose only. There can be no doubts about that purpose. He would not be educated, he would be brainwashed.
The Emperor is a monster who eats babies. The palace is filled with witches. He rapes children. There is a secret room in the palace just for castration because his wife is part snake.... He is not taught critical thinking skills, he is taught to be outraged.
This boy will not survive his 'purpose', but there will be virgins and ice cream waiting for him in the afterlife. Most important, he will redeem all the boys and girls who were sent before who failed. He will rescue their souls from the Emperor's hell fire. He will end their torment and they will all be his BFFs in the afterlife, being martyrs means special treatment, but what do you get for saving all the martyrs? That's got to be extra good, right?
You see why he can't tell the prolog. It would be crazy pants. He would undermine the sympathy of the reader.
In Medias Res
...before going on to attempt an assassination on the leader.
The technique of starting 'in the middle of the action' without explanation or set-up, is called in medias res.
IMO, the obvious place to start is not with the abdication of Prince So-and-so 72 years ago which led to a destablization of congress, etc. Start with the assassination, or as close to it as you can.
Maybe include his final good-bye/prep with his 'elder' who trained him (a person he will cross paths with again under very different circumstances, and knowing a lot more about the world). It's enough to understand it's not the protagonist's plan, but he is committed. We don't even need to know the plan.
His moral complication
the entire mission fails when he is forced to eliminate the leader
but cannot find himself to do so.
I suggest it is too early for pathos and a character turn, we're still in medias res and haven't explained his goal. Bring him to the position of being able to kill, but... reasons. (It is a public function, and he isn't making more martyrs.) Through show-don't-tell we know why he's there, we know who the target is, and we know he has some moral code.
Now he has to improvise. He gets even further into the palace maybe the Emperor's bedroom. This is his audition and it should be spectacular. Also by this point there is a suspicious lack of torture chambers and snake women –– maybe he has the gall to demand the Emperor reveal his true form.
(If the Emperor is a villain, this is him improvising without security. Charming and stalling and maneuvering, and figuring out who sent this kid to kill him.)
Some crazy-pants demands, after slipping past all the guards, is a mixed message. How does someone become this competent and this gullible...?
The reader learns he is not so competent just as the assassin glamor is running out. It gives the power back to the Emperor and crew, as his character needs to become vulnerable enough to (eventually) change.
Take the red pill
Then he is separated from his family, leaving him to believe that
there is no point in living.
If he is brainwashed and willing to die, I don't know how attached he would be to his parents (or they to him). I think he might double-down and resist deprogramming.
He believes they are lying, and might feel very righteous in his ability to stay mentally acute against their mind games. More crazy pants talk, as your story moves out of in medias res to introduce the larger cast and political intrigues.
If he's captured in the palace by the Emperor's personal guards, he can be disappeared and interrogated. Meanwhile, there's lots of pointing of fingers how he got in. You might have a difference of opinion within the court. Some feel he's a victim, some feel he's a propaganda opportunity –– but your villain with a plan has the power to hush it up.
There may be supporting characters who do simple jobs (cleaning, getting food), regular people with a sympathetic connection. They aren't all lizard people. His moral code still works, but he's conflicted.
Eventually he will cooperate or pretend to. If it needs to be convincing, bring in the expert de-brainwasher who declares him rehabilitated, for reasons of their own – like maybe they want to believe it.
Set the ball rolling
Finally getting back into the "main" part of the story, the boy uses
this opportunity to infiltrate and continue the mission.
The Emperor would definitely not allow some cult to send an assassin without having a meeting with that region's governor/ambassador, and sending out his own people to set things straight. The stakes are the Assassin's people can be wiped out at any time, but also it's a chance to show-don't-tell the political roles and power boundaries.
Assassin's people may have had regular skirmishes with that local governor, giving Assassin and Emperor an enemy in common. That could be the start of his new career.
There may be other conflicts triggered by Assassin's inciting incident. Whoever is in charge of palace security is now looking vulnerable. Any hard-liners against assassin cults are gloating right now. Opportunists will try to leverage the changing dynamics. Put lots of conflict around powerful people.
Assassin is a perfect 'orphan' and 'outsider' protagonist. The reader discovers the world as he discovers it.
Amoral, and less emo
I'm trying to liberate you from this emo depressed assassin, LOL.
If he is less 'internal' he can be morally ambiguous. Assassins are NOT relatable, and should not be framed as reluctant victims. An emo assassin just does not work, he could easily kill himself or get caught. He could pretend to be bad at his job.
You need an ambiguous protagonist, who is recovering from a life-long rug pull from the people he was ready to die for. Rather than breaking his character, I think it's better that he is trying to catch up on his knowledge so he understands who is going to do it next.
He is still crazy pants, but he is biding his time. Maybe he takes out a few lizard people while he works out who's who around the Emperor. Learns their habits. Applies his skills to a long plan.
He is also getting entangled in the intrigue himself, possibly breaking things that go against his code. Being an anti-hero who trusts no one, but allowing them to rely on him as a tool (a role he's familiar with).
