Before I start, let me just say that I haven't seen AoU yet, so don't put any spoilers in here. I'll update this when this changes.
With Spider-Man joining the MCU, there's a good chance that Spider-Man will be involved in the MCU's Infinity War. In the original Infinity War, the heroes battled monstrous doppelgangers of themselves, with most of them being destroyed at the end. One of the most notable exceptions was Spider-Man's Doppelganger, who went on to be in stories like Spirits of Venom and Maximum Carnage, and which was brought back recently in some of the first Carnage solo books. Though definitely not the deepest SM villain, I think if done right, he could look awesome on screen.
Now, since he's not the most intelligent villain, he probably couldn't carry a movie by himself. My idea is Spider-Man gets introduced in a solo MCU movie, but in the sequel, due to the actions of some mastermind villain, he is out of NYC and in a place where the media isn't. Doppelganger comes onto the scene and goes on a rampage. Since people in the MCU are still not that familiar with Spider-Man, they honestly that this is him mutated into a new form. Ultimately, Spidey comes back onto the scene and stops Doppelganger, but most people still believe that the creature's existence is somehow Spider-Man's fault. This could maybe feed into the public fear of super people that ultimately is expressed in anti-Inhuman bigotry (yeah, I know, I don't like it anymore than you do, but you have work with acre that you are given).
Another possibility is to include a mutation story somewhat based around the 90s Mutagenic Nightmare arc. It could be quite visceral if done right; if this is going to be a teenage Peter, imagine having him not only having to deal with the changes of puberty, but also with the problems of transforming into a six-armed monster. This could end with a showdown between Man-Spider and Doppelganger.
The ending could be very bittersweet since, even though the bad guys have been beaten, Spider-Man is hated and feared in the worst possible way as an absolute freak who needs to be stopped. Of course, he could clear his name in a later film, but this could be a fresh take on the Spider-Man film mythos, and I think that these ideas have at least some potential. Thoughts?