This card says after you win a military challenge in which Robb Stark participates, to choose and kill a character from the losing opponents side. Does this mean before or after the opponent chooses to kill one of their characters from the lost military challenge or is this the same character that has to be killed but having Robb Stark just gives me the added perk of being able to choose?

In other words, if my claim value is 2 and I use Robb Stark and win a military challenge does my opponent lose 2 characters from the initial challenge PLUS whatever character I choose for them to have to kill?

The faq will explain the timing structure to you in detail but the basic part is that all the framework procedures (like claim, un-opposed power and renown) all are resolved first, then all passives (like deadly), then you can play responses (like robbs) and when all responses are played all moribund cards are removed from play.

So in your claim 2 military challenge your opponent picks 2 characters to die for claim, then when you get to the responses part you can trigger Robb to chose and kill a 3rd target.