In these chapters, Augustine touches on two broad questions that no doubt his readers would be raising. (For a good historical background check out Collin Garbarino's notes on these chapters.)
The first is the question of whether Augustine's views on suicide mean that no one should ever kill under any circumstance whatsoever. Augustine's reply is clear: there are two circumstances in which one human being may take the life of another (the "human being" part is critical; God of course may--and does--kill any time he wishes with total justice). Augustine writes "However, there are some exceptions made by the divine authority to its own law, that men may not be put to death. These exceptions are of two kinds, being justified either by a general law, or by a special commission granted for a time to some individual."
In other words, the power to execute is delegated both under the general revelation (and here he means to the state in the form of war and the death penalty) and by special dispensation (to those individuals who took up the sword in God's name in the Old Testament.
Which is to say, according to Augustine unless you or I work for the state as a soldier or executioner, we have no right to take the life of another human being.
The second question is: what about the examples of suicides from history that seem to be clear examples of the right response by a great soul to a desperate situation?
In fact, Augustine argues that suicide is never a sign of greatness, but in fact is a sign of weakness and of the human lust for glory. We see this in the example of Cato the Younger, who was himself quite inconsistent on the value of suicide in the face of unstoppable evil. He himself took the "noble" path of suicide rather than live under Caesar, but at the same time advised his son to make his peace with the Dictator.
When we take our own lives, we are really making a last-ditch effort to seize whatever glory we can for ourselves. (And I don't know that Cato would necessarily have disagreed.) It is a final shout of a sinful will in rebellion against God, and as such no Christian should indulge in it. Of course, no Christian should indulge in any sin, but suicide is a fairly final one and so should especially be avoided.