I want to see Steve Rogers and Tony Stark as friends now. Steve talks to Tony about Tony's father, something that we didn't see in the movie. And maybe Stark tells Steve about what effect Captain America had on his father. They sort of bond over that and that could be the start of them actually talking more personally.
I wanna see how Thor interacts with the rest of the Avengers more, him respecting Captain America, etc. Thor is a demi god, and was in his youth a brat, but now he would fallow Captain America into Hel and back. In fact the whole team could trust Captain America, but there could still be a sort of power struggle because Iron Man and Steve are natural leaders and they both sort of try to lead the team.
I want to see Hawkeye looking to Captain America as a mentor of sorts, maybe being the sarcastic rebel type he was in the comics but really listening and learning the whole time, and I want to see Cap training Hawkeye.
I want to see, maybe, Jarvis the computer programmed buttler voice bicker with someone like the Hulk or Hawkeye and make snarky comments towards that Avenger. Clint could call Jarvis C3PO and Jarvis can complain about Hawkeye eating all of Mr. Stark's food. Their banter could also hide that Hawkeye and the artificial intelligence respect each other. (although I don't know how human Jarvis the program is really.)
I want to see Stark and Thor debating Science versus Magic. Stark trying to reason everything away and Thor trying to get him to just embrace it all.
In Earth's Mightiest he tells Thor he is a scientist and all this talk of Asgard and gods and magic doesn't make sense. and in the cartoon he tells Mr. Fantastic about Asgard and Thor but in scientific means and reasons it all away. Asgard could actually be this and Thor's hammer could actually be that, etc. Stark could also give the team a communications thing, like the ID cards are in the cartoon or ear buds like in Leverage. And then show how Thor reacts to that, add it to the whole how Stark reacts to Magic and how Thor reacts to technology.
Black Widow not becoming a love interest of Cap or Iron Man or Thor or Banner, maybe Hawkeye, but maybe not even him. I don't want her to just become about a love story, her character I mean.
Not really character thing here, but I want to see them living in the Avenger tower, this would force them to interact more.
I'd love to see Hawkeye play the part Spider-Man did in the Avengers comic, Thor leaps into action and slams a giant bad guy across the city with his hammer and Spider-Man (amazed) says, "Thats what its like being on a team with Thor!" It would show that even though they are still on the same team and friends they can still be wowed by his great godlike power.