The biggest hurdle would be establishing standing in order to bring such a case. The military would be the most affected, but they would have to follow the UCMJ regard illegal orders. Congress is having their consent ignored and could bring a suit, however a single representative or senator probably wouldn't qualify since the consent of the entire body is required and a dissenting minority can't file suits in protest. Normal citizens likely would have any suit they brought dismissed as lacking standing.
Congress also has more direct options that would probably be used if the felt strongly that the president violated the war powers resolution. There are multiple different bills they could pass to force the president to stop his actions, like defunding the military activity, changing the war powers resolution, or passing a resolution specifically disapproving of the presidents actions. They also would have the option to simply impeach the president if they felt he went to far.
The war powers resolution is meant to handle exceptional cases by giving the president leeway to handle them as he sees fit. Actions in response to violations of this resolution would be similarly exceptional, and any process followed would likely be made up at that time to handle that specific situation.