Well to start you have a couple options. Have the two universes been crossed for a while now or is this a new development? That answer dictates a lot in the kind of universe your characters will be growing up in.

If it has been a while then there are heroes and villians that would have had the chance to meet already. Things would be more "normal" IC and our characters would have a better grasp of who's who. This would be your more standard good vs. evil type setup. Leaves a lot of room for world building and preexisting setup.

Your other option is the crossover is new. You'll probably have a whole lot more chaos type of RP as we are seeing the reactions more on the fly when one character meets another that never have before period.

If we need a cause for something like this my goodness there are so many options. I can think of a that would work Well.