Constance is a woman with a mission: she wants her son Arthur to become king of England. She thinks it is his by right, because the tradition is that the crown passes from the father to his oldest son, and Constance's husband Geoffrey was John's older brother.
By rights, she thinks, the crown should have passed to Geoffrey and then to his son (Arthur)—not to John.
But just making this argument in words isn't going to get Constance anywhere; she needs some military muscle to back it all up. That's why Constance enlists the help of King Philip of France, as well as the Duke of Austria. And this is what Queen Eleanor is talking about in Act I, Scene 1, when she tells her son John that Constance is behind all of his troubles (at lines 31-38).
Even though the social rules of the day mean that Constance doesn't have that much direct political power, the fact that she basically gets the whole plot going reveals her as a strong woman who gets things done by pure gumption.
Unfortunately, gumption only gets you so far. Constance's major problem is that she's limited by the people she has to work with. For one thing, it seems pretty clear that King Philip and Austria are only interested in helping her because they think they can manipulate Arthur for their own purposes. King Philip, especially, seems to think that Arthur is his ticket to getting control of England.
Thus, even if their war against King John had been successful, it isn't clear how much Constance and Arthur would gain from it. Also, Constance seems to forget that, when your allies are helping you for personal gain, they're likely to abandon you when an opportunity for more gain comes along.
After all, when King Philip sees that he can secure a good deal for his son Louis by marrying him off to Blanche, King John's niece, he doesn't think twice about leaving Constance and Arthur out in the cold. When Constance learns about this, she doesn't back down: she lashes out bitterly at King Philip. She finds out pretty quickly, however, that pretty much no one is to be trusted: King Philip doesn't change his mind.
The most Constance can do at this point is join the chorus of voices encouraging King Philips to break his alliance with King John, in keeping with Cardinal Pandolf's instructions. But by sheer dumb luck, this turns out terribly for Constance: in the ensuing battle with John, Arthur gets captured—which sets in motion the chain of events that ultimately leads to his death.
Immediately after Arthur has been captured, Constance becomes tormented with grief. Her behavior is erratic and self-destructive; she even makes lewd, disgusting remarks about death. And yet, in a classic twist that shows the depth of her character, she argues that she isn't crazy; in a situation of such horrible grief, she argues, it would be crazy not to act crazy. She eloquently puts in their place the men who try to calm her down, arguing that they can't possibly understand what she is going through.
After this climactic scene, Constance basically disappears from the action. Towards the end of the play, we learn through a messenger that she has died. Could it be she has died of grief?