vaisravana's Favorite

Since the narration isn't linear, you just have to separate the story in two timelines:
- The first one, the actual 'beginning' of the story, is shown in the flashbacks at the end (black and white scenes). Basically, Law's parents manipulated him into getting the Brown's favors through Jesse to get privileges in the high society, and he willingly did all that because he thought he wasn't his father's legitimate son - out of a sense of guilt and fear of being thrown away, I suppose. His friendship with Jesse is super strained at first, Jesse's personality considered, and that puts Law under a lot of pressure and anxiety, to the point where he starts hating Jesse as much as he loves him.
Anyway, Jesse says that about wishing everyone would forget his condition after the incident with the maids, and Law is so devoted he takes it seriously and creates Moritat based on that projector-thing Jesse made when they were younger. Jesse is disgusted by Moritat and its mind-altering purposes; his mom loves it, seeing Law as a potential cash cow and leashed 'dog' to the Browns. After Law learns he's his parents' legitimate son, he resents everything they made him go through to become Jesse's friend and is fairly content when they die (probably killed by the Browns, so Law would be farther within their range of control).
Jesse cares about Law enough to hate the idea of his own parents controlling him and bans him from the mansion. Later on in the story, the media comes to know about Moritat being used on the staff in the Brown manor through Rosalie, who worked there as a maid, and Jesse takes responsibility for the whole mess, so he hides from the public eye and goes to an ordinary university. The scene where Law appears and Jesse is sitting on a bench happens years later, after all that; Law gives Jesse the tea that's used to ease the person into Moritat and puts both of them into the virtual world he's created, still longing to be Jesse's one and only.

- The second timeline starts after everything I've described above, from the first chapter until Jesse chooses to die with Law and starts bleeding from the head. Everything that happened in those chapters didn't actually, well, happen. Law programmed a bunch of situations that that would toy with Jesse's mental state over and over to see how Jesse would react, if Jesse would trust him and stay with him if he hadn't had everything he could wish for in the palm of his hands. And also, I presume, as a means of letting out all his conflicting feelings about Jesse - his love, his heartbreak, his anger and his agony. The simulation fails multiple times (for example: the scene at the beginning, where Law pushes Jesse off the cliff, is the failed ending of the previous simulation), until Jesse finally gets around his feelings for Law and chooses to die with him; that's when Law turns off Moritat, because that's exactly what he always wanted from Jesse: to be needed. He pretends not to love Jesse anymore when they wake up to real life, though - and that's just one day after meeting again at the university, by the way; it's only Law who feels like it went on for longer, since he was the 'administrator' in control of the simulations -, because he can't deal with the idea of Jesse loving him.
At the end, after meeting Rosalie, Jesse goes after Law and sees him using the projector-thing from their childhood. It's meant to project a certain image based on your feelings towards the subject of the image, and that's how Jesse knows instantly that Law still loves him: the image of the young Jesse that appears is the same as all those years ago. If his feelings had changed, it would have looked different. Jesse snaps him out of it and they both, finally, come clear about their feelings.
Hope this sorts some of the story out!