Portia reached Earth through Bruks, as you say, through deliberate infection by the Bicamerals. It reached the Icarus array through, essentially, a hack.
We do get a bit of information on Portia's arrival on Icarus in the text itself:
(...) Which implies that it’s optimized for telematter dispersal—although if it isn’t using our native protocols, how it hijacks the stream in the first place is beyond me.”
“Oh they figured that out couple days ago,” Sengupta told him.
“Know how when you pack a layer of ball bearings into the bottom of a crate and the second layer fits into the bumps andvalleys laid down by the first and the third fits onto the second so it all comes down to the first layer, first layer determines all the turtles all the way up, right?”
“Like that. ’Cept the ball bearings are atoms.”
“You’re shitting me.”
“Yah because I got nothing better to do than play tricks on roaches.”
“But—that’s like laying down a set of wheels and expecting it to act as a template for a car.”
“More like laying down a set of tread marks and expecting it to act as a template for a car.”
The author sort of handwaves (under intelligence-far-greater-than-humanity) exactly how Icarus is convinced to send that first layer down, but considering Portia shows evidence of knowing what happened on Theseus (or at least, of what the narration said happened), it stands to reason that Theseus, or part of it, was captured and while in enemy hands their own telematter interactions were able to send something back along the supposedly one-way stream.
Portia got into Bruks through deliberate infection by the Bicameral order, who had it as part of their incomprehensible long-range plans (unfortunately, their plans were driven off course by Valerie's own long-term plans). It was one of the reasons they brought him along (as a baseline, there'd be less interactions to worry about).
The Portia infection was mistaken for cancer at first.
He’s gone again, he thought, but not entirely: Moore dropped that distant gaze from Brüks’s face, took his hand, pointed to a freckle there that Brüks hadn’t noticed before.
“Another tumor,” Brüks said, and Moore nodded distantly:
“The wrong kind.”
It's perhaps notable that the freckle was on his hand, which might be the easiest way to infect someone without their knowledge. I can't find a specific passage where an infection clearly took place (though, considering it was a surprise to the viewpoint character, it presumably is either extremely hard to find or absent entirely), but they did take samples of Portia and ensure that he got them, including one which was delivered by hand:
“GOT SOMETHING FOR you.”
It was a white plastic clamshell, about the size and shape to hold a set of antique eyeglasses. Lianna had fabbed a bright green bow and stuck it to the top.
Brüks eyed it suspiciously. “What is it?
“The Face of God,” she declared, and then—deflated by the look he shot at her, “That’s kind of what the hive’s calling it, anyway. Piece of your slime mold.” She held it out with a flourish. “If Muhammad can’t come to the sample…”
“Thanks.” He took the offering (try as he might, he couldn’t keep from smiling), and set it on the table next to dessert.
“They thought you’d like to take a shot at, you know. Seeing what makes it tick.”
However, Watts himself makes the Bicams infecting Bruks explicit in this response on an AMA he did:
Picky point, though: "Bicam cancer" is probably a misnomer. What the Bicams infected Bruks with (once they'd derived sufficient understanding of what they were playing with) was Portia. It was Valerie who hacked it to her own ends, not the Bicams. And while it superficially might have manifested in a tumor-like way (along with other real cancers that sprang up due to their proximity to the sun), Portia was never a cancer.
Of course, even though I designed the plot in such a way that the Bicams didn't do any tweaking on their own, I never stated as much in the novel, so it's not canon. I suppose, if I ever revisit this world and I've got some reason to change that, I've got the wiggle room.
The tweaking referred to is what Valerie infected him with in their last scene together, which was supposed to be a "patch" on the Portia already inside him.
As far as I can tell, it wasn't implied that Portia reached Icarus through Siri's transmissions directly (Icarus wasn't set up as a communications relay, just a telematter stream, so it would have made more sense to attack it by manipulating the existing protocols for that rather than a long-winded narrative message nobody there would be expected to hear), although it's still possible that the Siri transmissions were forged and also come from hijacking Theseus, much like the telematter hack was. It just may have been a Trojan horse in another way, to sow dissent on Earth, etc. But it at present is still unconfirmed that the transmission was false, we'll likely learn for sure in Watt's proposed final installment (currently under the working title of Omniscience but not yet published).