Traditionally, the "Anticipation Stage" of tragedy finds the hero somehow incomplete and in need of fulfillment. In one way, this doesn't apply to King John, because he already possesses the territories and the crown that he is going to fight King Philip over. That said, because King Philip is threatening to take that stuff away and give it to John's nephew Arthur, it's almost as if John doesn't actually have that stuff, since he doesn't have it securely.
The "Dream Stage" of tragedy usually happens when the hero gets committed to his course of action; it's the point of no return. We'd say that once King John goes to France and fights a bloody battle with his adversary, he's pretty darn committed to his quest.
King John makes a marriage-alliance with King Philip, only to have it broken up by the meddling of Cardinal Pandolf. When King John isn't able to win a victory through military combat, the next best option for securing his kingdom seems to be forging an alliance with King Philip—by having his own niece Blanche marry Philip's son Louis. (The old "if you can't beat 'em, join 'em strategy.")
But trouble rears its head right after the wedding vows have been solemnized, when Cardinal Pandolf shows up as an emissary from the Pope. The Pope has an ongoing beef with John about the selection of the next Archbishop of Canterbury. When John refuses to back down, Pandolf turns against him and incites King Philip to turn against him, too. As war breaks out again, the tidy solution seems to be coming undone.
Things definitely start falling apart for King John at this point in the plot. Nothing seems to go right for him. When he orders Arthur killed, his henchman Hubert fails to carry out the order—but he still tells King John that he did it. The news that Arthur is dead fails to help King John out; instead, it turns his nobles against him.
Then King John finds out Arthur is alive, but he fails to get any benefit from it, because in no time, Arthur dies for real. When the nobles find Arthur's body, they turn decisively against John and ally themselves with Louis, who is coincidentally invading England at the time. You can tell that John is flaking out because he entrusts his entire military command to the Bastard, a relative nobody, instead of taking charge himself, as a king should.
Destruction or Death Wish Stage
John leaves the battle because he is feeling sick. It turns out that he's poisoned by some monk. He prays for a speedy death... and he dies. On the one hand, it's a little hard to say that John has a "death wish" at this point, because he does get poisoned by somebody else (some random monk) and not by himself. On the other hand, John's behavior is so flaky at this point and shows such a lack of responsibility that he might as well have taken his own life.
In any case, once he's has been poisoned, the suffering is so intense that he wishes to die. And soon enough, he does.