I haven't been able to respond due to lack of an account, but I'd like to weigh my opinion on this subject.
If a Venom film is to work in this new Spider-Man universe, the first thing in my mind would be to avoid the alien plot. The symbiote suit could be tied with Oscorp, as there is already a foundation for such experiments in this new universe. Oscorp has already been established as a factory for some of the most evolutionary inventions and scientific devices on the planet. They produce Parker's webbing, provided his powers, perform genetic experiments on all sorts of creatures, and even had the questionable Ganali Device.
As a result, it would only seem fitting if Venom's origin stems from an experiment at Oscorp Industries. They could borrow elements from the Ultimate universe, as director Marc Webb has already done with the first film, and perhaps involve the genetically-altered spiders in another DNA experiment. If Peter Parker inherited his powers from the spiders, one could easily also adopt those powers if they could harness the power of the spiders.
I would propose the following as a potential idea:
Richard Parker's experiments, as Doctor Connors stated in the first film, dealt with genetically-altered spiders that would 'change the lives of millions.' Clearly, this meant spider DNA would have been used initially; when this was destroyed, Connors started over and used reptilian DNA instead. The first film has already established a workable canon that could involve a substance or suit involving spider DNA.
Let's suppose Richard Parker's research resurfaces in the future; this could perhaps occur in a future Spider-Man film, as the origin story is continually being uncovered. In that case, perhaps such research would be used to develop an animated, living suit that would cure disease and prolong life as well as youth by secreting its genetically-altered spider-like DNA into the bloodstream and enhancing the recipient's bodily functions. The suit would have to be worn at all times and could only be unbound by select Oscorp sonar devices. It would, theoretically, be used for medical purposes, though Norman Osborn could very well have been tampering with the design, transforming it into a biological weapon.
As with most symbiotic organisms, the substance would be designed with good intentions to seek disease and toxins. The suit, however, would poison the user's bloodstream, causing immense pain and partial psychosis, as well as embedding spider DNA in the bloodstream permanently. The suit would also bond with the person's skin, liquifying and regenerating if damaged. Its DNA would combine with the wearer's DNA after a certain amount of time, eventually allow the user to control and manipulate the suit with trial and error. The suit itself would be permanently bonded to the wearer's body, though the user could control whether the liquid extends on to his face. Its DNA would cause the user to develop a second, more predatory, spider-like personality, resulting in Venom's dual personality. The suit would be manifesting itself in the user's personality via its DNA.
With that, we could have the basis for Venom. The wearer would inherit an upgraded version of Spider-Man's powers as a result of the same spider DNA, though in a "more-controlled" fashion dependent on the suit.