International Law is a very fuzzy thing. As you've found out, there are international courts. Specifically, the International Criminal Court (ICC) in the Netherlands is the body that exists to hold trials for individuals accused of war crimes.
When people talk about arresting and holding a trial for Kony at the international level, they mean the ICC. The ICC's prosecutors have indicted Kony for war crimes and crimes - they want the guy brought in and put on trial.
But the ICC is purely a judiciary body. It has judges. It has lawyers. It doesn't have international police jungle paratrooper commando squads. The ICC can only hold trials for people turned over to the ICC. Someone else has to catch them and ship them over.
That's probably not going to be the US, because the US does not support the ICC. The US isn't part of the treaties and agreements that back up the court, because the US doesn't want to have American soldiers dragged before the ICC on war crimes charges.
So the chances of US soldiers going out and grabbing Kony on behalf of the International Criminal Court...no, not gonna happen. Not like that, anyway.