If you complete the following tasks you will be able to get an assist from the corresponding character.
Big Wheel: After beating Tinkerer keep selecting the 'bad' choice when asked by computer.
Black Cat: Select to save Black Cat when you fight her (pick the 1st choice when asked by Smythe)
Hobgoblin: Before fighting Kraven keep selecting the 'bad' choice (the 2nd/3rd choice).
Jessica Drew: Before you fight with Jessica Drew Select to help S.H.I.E.L.D crew from the symbiote.
Luke Cage: Select the 'bad' choice (2nd choice) when you asked by a man who says he can divert the train containing symbiote.
Shocker: Choose to free Shocker when you have defeated him.
Solidarity: When you have defeated Shocker take him to jail.
Storm: When you have defeated Tinkerer keep selecting the 'good' answer when asked by computer.
Random Assist: When you beat any level or buy the ultra rare power ups you will be able to get a random assist from the following characters - Mysterio, Night Crawler, Wolverine, Galactus, Hypno Hustler Ctylok.