Look to the God of War games for some inspiration if you need it. Kratos is always doing ridiculously epic things (especially in the second and third games) in the name of war. He actually is the God of War for a bit!
Other than that, I'd say the most important thing about an Exarch of Kord is that they aren't just about battle. They are actually physically drawn toward it. They cannot avoid battle, because that is what they are. So, if he wants to truly become the exarch of Kord, he needs to find the battle of his time and take part in it, regardless of what the stakes are.
If that means he has to visit another plane of existance, that's what he'd do (and since it is epic, that would fit thematically). Maybe he actually has to take part in the Blood War (not technically part of 4E anymore, I don't think, but still a cool idea) - and not just participate, but actively change the flow of the millienia old war.
Maybe he has to skip around to multiple different worlds looking for the ultimate battle to take part in to prove his worth to Kord.
Those are a few thoughts, just off the top of my head.