I actually think there are a few ways to make a Venom movie work with this re-booted Spider-Man franchises, especially considering how The Amazing Spider-Man seems to be drawing more from Ultimate Spider-Man.
Have Eddie Brock be a research assistant and part-time photographer in Los Angeles - he can be either small and scrawny or bigger, depending on what they decide - with an ego and a temperment. He's working on curing cancer using genetic research, but is frustrated by failure, and can't get recognised for his photgraphy talent. Maybe he's a bit of a creep and a letch around women, too.
Then, someone gives him a blood sample to help his research, obtained from an outside source (Dr. Conners would be good, but that would depend on how The Amazing Spider-Man ends. Maybe Osborn; either way, Eddie's superior). The sample reacts with Eddie's black substance, turning it to vicious, malformed webbing that grows and pulsates. Eddie is excited by the reaction, but this turns to horror when it latches onto him and turns him into a monstrous, tentacle-spewing creature.
Out of control, and struggling to regain his sanity, this creatures goes on a rampage. Now, with a PG-13 rating, there are plenty of ways to show or imply Venom killing or hurting people - look at the Doc Ock scene in Spider-Man 2. Just quickly cut away, show Venom hurling and clutching at people, that sort of thing. Eventually, after fighting with the police, Eddie manages to wrestle control and webslings to a rooftop, where he collapses, the suit slimming down to a more classic Venom look.
Eddie roots out the guy who gave him the blood and harasses him for answers; the doctor says the sample came from New York and was Spider-Man's blood. Eddie realises that he needs more of the blood to gain complete control of Venom, and leaves. The doctor, seeing the power Venom has given Eddie, then exposes himself to his own strain of the sample, becoming Carnage, and begin his own massacre, for which the press blame Venom due to the similar nature of the attacks.
Venom and Carnage confront each other in the final act, with Venom triumphing. He destroys the creature, leaving the doctor at his mercy. Eddie exposits about how he feels drawn to New York and to Spider-Man and that the key to his salvation lies there, and begins to leave, then kills the doctor before he does so. The last shot of the film is Venom making his way to New York.
The Carnage part isn't that great, maybe just fighting military types would be better, but the aim of the film is to establish Eddie/Venom as an anti-hero, the kind of guy who kills to reach his goal and is obsessed with the power he has, even though it frightens him and is often beyond his control. Introducing a duality between Eddie and Venom could be good, with the two finally reconciling and becoming "one" at the conclusion.
This could be a good setup for a future Spider-Man film, wherein Spider-Man fights with venom, and the two ultimately team-up to face either Carnage (if not used in Venom) or another common foe. The only draw back as far as I see is we don't get to see Black-Suit Spider-Man, but if they're serious about making Venom I would prefer to leave Spider-Man out of the film entirely except for name-drops and maybe TV footage, and save the two meeting for their joint film.