In "The Archbishop and the Chambermaid," William Godwin poses an imaginary case in which someone has to choose between saving the life of Fenelon, the illustrious Archbishop of Cambray, and the life of a chambermaid. What does Godwin believe is the right way for a strict utilitarian to act? Do you think that Godwin's moral approach is the correct one? What is a potential problem raised by this case?

Interesting question! What we're dealing with here is act utilitarianism versus rule utilitarianism. Basically, Godwin's approach can be said to be a strictly utilitarian one, based on an evaluation of individual actions. We would call this approach act utilitarianism, where actions are evaluated based on their consequences. In act utilitarianism, the actions that result in the greatest good are the right ones.
On the other hand, rule utilitarians evaluate individual actions based on an absolute moral code. To rule utilitarians, actions that adhere to a prescribed moral code constitute right actions that will result in the greatest good. Godwin's approach is strictly utilitarian in the sense that he is only interested in the consequences of the action (act utilitarianism). He argues that saving the Archbishop would result in the greatest good because Fenelon is better positioned to save society from moral decay: 

Of consequence that life ought to be preferred which will be most conducive to the general good.  In saving the life of Fenelon, suppose at the moment when he was conceiving the project of his immortal Telemachus, I should be promoting the benefit of thousands, who have been cured by the perusal of it of some error, vice and consequent unhappiness. Nay, my benefit would extend farther than this, for every individual thus cured has become a better member of society, and has contributed in his turn to the happiness, the information and improvement of others.

Here, Godwin is referring to François de Salignac de La Mothe-Fénelon, the Archbishop of Cambrai. His most famous work, Les Aventures de Telemaque, is a critique of monarchical excess and an exhortation to embrace the principles of egalitarianism. For his liberal views, Fenelon was exiled from Versailles by King Louis XIV of France. However, Fenelon's message of spiritual enlightenment and social egalitarianism was later embraced by John-Jacques Rousseau. Although Rousseau and Fenelon differed in their focus on spiritual matters, both deeply supported the principles of egalitarianism.
Godwin's approach is antithetical to Kant's, for example. Kant would argue that speaking a lie is always wrong, even if it saves someone's life. His rule utilitarianism contrasts with Godwin's, who would argue that rule-breaking is not intrinsically an evil act, as it can sometimes result in positive consequences. 
So, a potential problem with act utilitarianism is that it conflicts with widely-revered moral codes. Act utilitarians like Godwin would submit to the killing of an innocent person if it meant that the lives of countless others could be preserved. Rule utilitarians argue that such an action constitutes murder. They contend that act utilitarianism undermines trust between individuals in society: If people are allowed to lie, cheat, and kill at will, society will be destabilized.
Rule utilitarians disagree with Godwin, who argues that principles like gratitude "which has so often been the theme of the moralist and the poet, is no part either of justice or virtue." In other words, Godwin believes that the rightness of an action must be determined by what it achieves, not by its adherence to an entrenched moral code. In arguing thus, act utilitarians fail to take into consideration the moral legitimacy of self-interest. 
In your opinion, should the maid's interest in preserving her life be considered equal to that of the Archbishop's? Your answer will determine whether or not you agree with Godwin's moral approach.
Source:  Men and Citizens: A Study of Rousseau's Social Theory by Judith N. Shklar