This page contains SPOILERS for Broken Minds. Proceed with caution.
Junji Uzumaki's motive for being The Orphan is never truly confirmed in the game, though a few theories pop up.
Noa's Hints[]
In some of her confessions on the Noa is Guilty route, Noa reveals some things about Uzumaki, which could point to possible motives for the Uzumaki is Guilty route.
See, he was orphaned at a young age, and he got to be independent early. That helped him make his own decisions, without parents in the way. And having no parents produced pity in his friends.
He would buy me these pet rabbits, and tell me... These are your pets now. You must take care of them. Then he would kill them.
Since these things are true no matter what route you're playing, this reveals that Uzumaki was sadistic at best, and also believed having no parents was better than having them.
He would say things like "I'm doing a horrible job treating you, aren't I?" or "I'm sorry I look so ugly today." It was like he was fishing for compliments. Also, I know it's a therapist's job to empathize with clients. But he seemed disinterested in hearing me talk, and instead went on long tangents.
This seems to point to Narcissistic Personality Disorder.
Ume's Bad Theory[]
Ume has her own ideas about Uzumaki, which are criticized by Noa.
Uzumaki saw all of his patients as surrogate parents. Even the little girl, Amane. But that changed with you. For some reason, he began to think of you as his child.
In Ume's theory, Uzumaki believes that his patients are like his parents, and that he is still a child. She points to "playful" elements of his office, like the indoor pond (and presumably the sunken cushion-filled section), as evidence of his child-like mindset. She claims that Uzumaki killed Noa's parents so that he could "relate" to Noa as only orphans can. But then he killed himself because he started to see himself as Noa's parent instead of her child.
This theory is full of holes, for instance, if Uzumaki knew he was an orphan, how could he also think that Noa was his parent?
Ume's Better Theory[]
In the final deduction section of the Uzumaki is Guilty route, Ume revises her theory to make a little more sense. However, it is slightly crimped by the fact that Noa lies about Uzumaki, pretending that he isn't an orphan after all.
What if the letter didn't mean Uzumaki specifically? What if it really meant "Now you're one too, like this friend I know"?
Ume's new theory is that Uzumaki wanted to make an orphan friend for Amane, because he mistakenly thought that Amane was an orphan, and, in his twisted way, believed Noa had to be an orphan in order to be friends with Amane.
He killed himself because he was afraid of being caught.
Uzumaki's Hints[]
DEEP SPOILER ALERT! You've been warned.
Though Junji Uzumaki only has a few lines in the game, some of what he says points to Ume's better theory being mostly right.
In the Uzumaki is Guilty True Ending, he mumbles something about Amane needing a friend, so that, at least, is confirmed.
His logic appears to be that being an orphan means you are pitied, and being pitied means you will make friends, something he told Noa. So, on the route where he kills Noa's parents, it can be assumed that he did so so that Amane will pity Noa and want to be her friend. Remember, Noa has a harder time getting other people to be friends with her than she does being friends with other people, so in order for their friendship to work, Amane needs to want to be friends with Noa.
Uzumaki appears to be cognizant enough to be afraid of being caught at murdering Noa's parents, which is why he kills Noa in the true ending, and kills himself in the Uzumaki is Guilty route.