The story to the multiplatform game Ultimate Spider-Man was written by Marvel comics scribe, Brian Michael Bendis. It takes place in the "Ultimate Spider-Man" line of comic books, which is a modern re-imagining and re-telling of classic Spider-Man lore. The events of the game are considered canon, therefore they will affect events in the comics. At the same time, threads and seeds are planted in the game that are unresolved, but may be used in the comic at a later time.
The following is history of the characters involved in the main storyline that take place before the game, specifically the origin of Venom. It is assumed you know that Peter Parker is Spider-Man, and that Spider-Man does whatever a spider can.
Peter Parker and Eddie Brock Jr. were childhood friends. Their fathers were scientists that worked closely together, and died in the same plane crash. When 15-year-old Peter Parker reconnects with Eddie after all these years, Eddie shows him a large vial of black liquid, the last project their fathers were working on: a suit that could potentially cure cancer, codenamed Venom.
When alone, Peter Parker takes some and accidentally exposes his skin to the liquid. It covers him and becomes a new, black suit. It enhances his abilities, and even allows him to take bullets. However, before long, he finds that it is too difficult for him to control. So he rejects the suit, takes the vial, and tries to destroy it.
Regardless, it ends up on an enraged Eddie Brock Jr., who feels betrayed by Peter's actions (trying to destroy their fathers work), so they fight in the rain. Eddie is not in control of the suit, and is more or less an out of control animal. It ends with the police shooting at Venom, as a stray electrical line shocks the symbiote off him. Nothing is seen of Venom or Eddie again.
In the game
Eddie Brock awakens and isn't sure what's going on. The Venom suit takes control before long, and he's forced to feed on civilians in order to sustain his health. He is capable of huge 100 foot jumps, super strength, and various acts of tentacle goodness. Or badness. In his first night out, he encounters the X-Men mainstay, Wolverine, and pounds him to unconsciousness.
Meanwhile, Spider-Man stops the supervillain Rhino from levelling a whole block just before his Trigonometry final. After the fight, Peter finds that his skull is aching terribly, and assumes it's from his most recent fight. As it turns out, a very lost looking Eddie is walking around aimlessly nearby. They don't see each other.
By night, the Venom suit takes control again, and this time he's being hunted by mercenaries with guns and jeeps and lasers. They are the Wild Pack, a group of mercenaries led by Silver Sable. She watches from afar as Venom practically eats and/or beats up an army of them. He escapes, much to the disappoint of Silver Sable, who was hired by Trask Industries to recapture Venom.
Before long, Venom shows up at the Museum, and Spider-Man arrives to set him straight. As it turns out, whenever Spider-Man goes near Venom, his Spidey-sense is so overwhelmed that it causes intense pain. As a result, Spidey is forced to hit & run. The fight is ended when Silver Sable and her men show up, zap Venom a bunch of times, and take him away in their gigantic hovercraft. Spider-Man is left bewildered as to who they are, and what just happened.
Venom wakes up in a containment cell at Trask Industries. He is confronted by Bolivar Trask, CEO, and his assistant, Adrian Toomes, a scientist who worked under both Edward Brock Sr. and Peter's father. They captured him because they owned the Venom suit, and it's worth billions of dollars. They tell him the suit is consuming him, and before long, he will die. Trask tells him that he might as well be their test subject, since if he dies, they'll simply find a new guinea pig. Eddie reluctantly agrees and asks, "So what do you want me to do?"
Test the suits abilities, for one. The supervillain Electro is paid to lead Venom on a chase, and then duke it out. Venom keeps up, and they begin to battle in Times Square, where Peter Parker happens to be working at the Daily Bugle. He comes out as Spider-Man, but is quickly knocked cold by Electro. Venom manages to defeat Electro before he can kill Spider-Man. Apparently, Venom is now obsessed with making sure HE is the one who eats/kills Spider-Man. Shield appears to clean up the mess, taking in Electro, but failing to apprehend Venom. Pete is angry that they can't keep their supervillains locked up.
Suddenly, a guy in a flying armored suit called Beetle appears in New York City. It turns out he's an agent of Latveria, the country that Dr. Doom rules. His objectives are odd and varied, from freeing the Green Goblin from Shield control to taking a vial in a laboratory labeled, "Sandman." Spider-Man confronts him, and is blasted. They fight at the top of a skyscraper, but Beetle leaves before he can be beaten.
Elsewhere, Trask and Toomes learn that the suit was once worn by Peter Parker. They also learn from Eddie that he only felt powerful and in control of the suit when Peter Parker was near. This leads them to believe that some of the suit is in Peter's blood, and that the suit is trying to bond with him again. They also know that Peter is Spider-Man. So, they go out and try to find him. They have Silver Sable put Eddie in a hovercraft, but he turns into Venom and escapes. Disappointed, but Silver Sable knows that she can still go after Peter instead.
The escaped Green Goblin is angry, and has a score to settle with Peter/Spider-Man. So, they fight. Peter wins, and Shield does clean-up again, taking the Goblin back into their custody. As for Eddie, The Beetle pursues Venom, but fails. Venom knocks him out in a wearhouse and leaves him and his gear smoking.
The next day, Peter finds himself being abducted and pumped full of tranquilizers by Silver Sable. They fight on the Queensboro bridge, but it causes accidents, and they both stop to save civilians. But wait! Here comes Venom! In his rage, he takes Silver Sable, and Peter stands back, still suffering from a few dozen shots of tranquilizer. "Are you just gonna stand there?!" asks a bystander. "Thinking about it," replies Spider-Man, before reluctantly chasing Venom, since it's the right thing to do.
They fight, and Spider-Man wins. However, the tranquilizers finally take their toll, and he falls unconscious soon after. Silver Sable wakes up to find both Eddie Brock and Peter Parker knocked out on the ground. Score. She and her team takes them to Task Industries. Venom wakes up, and his containment cell is mysteriously opened. Out the door comes...Carnage. Or at least something that looks a lot like him.
They fight until Venom wins. Venom absorbs him into his suit and spits out Peter Parker. As it turns out, Toomes injected Peter with something as an expirement after capture. This turns him into "Carnage", and in his out of control state, breaks out and wrecks the labs. He manages to free Venom, which is how their fight began.
The absorbing of the Carnage suit changes Venom. He now sports the famous white emblem on his chest, and is finally in full control of the suit, since it's technically "bonded" with Peter Parker, or the remains of the suit that was in Peter. The first thing he does is get revenge on Bolivar Trask, who tries to escape in a Helicopter. Eddie is enraged at his treatment, containment like an animal, and for his belief that he is the reason their fathers died in that plane crash. Spider-Man fights off Venom, but he, as usual, escapes. Bolivar Trask is saved, SHIELD cleans up, and he gets his hands on a file that has all the information his & Eddie's parents.
Reading through it, he learns that the plane crash wasn't a staged crash caused by a malicious and greedy Bolivar Trask. Eddie Brock Senior tried the suit on in the plane, which went out of control, and destroyed the plane. Peter is forced to resign to the fact that Venom is out there somewhere, and they'll probably have to fight again someday.
While he may have stopped him from killing Trask that rainy night, Eddie shows up at Trasks prison to confront him. He is even angrier that after all of his crimes, all he gets is 3 years in a "golf prison."
So he eats him. Presumably.
The story ends with Eddie loose on the city but finally in control of his powers, and Peter alive after another crazy adventure.