While Spider-Man is known for being a friendly and approachable hero, the alien symbiotes that eventually came to be known as Venom and Carnage are anything but. Although the villainous aspects of the characters were carried over into their Ultimate Universe iterations, many other features of the characters were altered, as was par of the course for the new reality.
Instead of an having an extraterrestrial origin, the Ultimate symbiote was the result of a scientific experiment conducted by Peter Parker's father. This led to very different versions of both Eddie Brock and Carnage. Here's how the Ultimate version of Spider-Man gained a black suit without any sort of extraterrestrial interference.
Ultimate Eddie Brock
Eddie Brock in the Ultimate Universe is actually Eddie Brock, Jr., and has several differences that set him apart from his mainstream counterpart. Brock is still a journalist in this incarnation, but he's also a lab assistant for Curt Connors. This, along with his more gangly appearance, was a huge contrast to his body-builder physique in the 616 Universe, though his interest in science inspired his similar incarnation in the Spectacular Spider-Man cartoon.
This version of Eddie wasn't Peter Parker's rival, but instead his childhood friend who reunited with him after Peter became Spider-Man. Their scientist fathers worked together previously, which resulted in the creation of what would become the Venom symbiote. This gooey suit was created by Eddie Brock, Sr. in hopes of curing cancer, and was engineered to work with Richard Parker's DNA in early testing. Because the suit was engineered for him, the elder Parker was able to easily control it while it protected him from injury.
If someone whom the suit was not meant to bond with does so, however, it will cause immediate damage to their body. To counteract this, the suit will have to consume immense amounts of nutrients, gained mainly through the consumption of organic flesh. It also leaves trace amounts of itself in a host's bloodstream, which can attract larger amounts of the symbiote. While the mainstream symbiotes are weak to sonic frequencies, this version of the suit is instead weak to electricity, with large amounts of it killing the suit.
Ultimate Venom
Peter later discovered that Bolivar Trask intended to use Richard Parker and Eddie, Sr.'s research for nefarious purposes, and stole the suit before accidentally bonding with it. The suit worked due to being based off of his father's DNA, and it greatly enhanced Peter's already superhuman strength and agility, as well as giving him organic webbing. When it began taking control of his personality, however, he destroyed it via electricity.
Eddie would later take a second sample of the suit and bond with it himself, becoming Venom. This form was initially just a more muscular version of Spider-Man, much like how the classic Venom looked when he was first drawn by Todd McFarlane. Eventually, however, he grew fangs and a more monstrous appearance like in the mainstream universe. After bonding with Peter and bonding back to Eddie afterward, Venom gained an even bulkier appearance.
This led to a saga involving Trask and Beetle trying to capture Brock and recover the symbiote, with small parts of the suit also bonding with Gwen Stacy, turning her into Carnage. Venom would defeat his crimson twin and absorb the Carnage symbiote, becoming even more powerful in the process. After escaping from Spider-Man and the now powerless Gwen, he was subsequently captured by Beetle and taken to Latveria.
This was the last appearance of Eddie Brock as Venom in the Ultimate Universe, and it's unknown what became of him afterward. This take on the character was certainly different from the mainstream version, and appeared in the Ultimate Spider-Man videogame. It also inspired the symbiote and Eddie Brock's portrayal in both Spectacular Spider-Man and the Ultimate Spider-Man cartoon, showing the longevity of a more earthbound symbiote.