According to Aristotle, what is the relationship between a thing's excellence and its good?

The first thing to realize is that, in Greek, a thing's excellence (arētē) and the good (to agathon) are closely linked in meaning. However, in his function argument (Nicomachean Ethics Book I, Chapter VII), Aristotle argues that the excellence for something is for it to perform its function (ergon) well. I take it that this is the argument to which you are referring. To summarize: The function argument states...

The first thing to realize is that, in Greek, a thing's excellence (arētē) and the good (to agathon) are closely linked in meaning. However, in his function argument (Nicomachean Ethics Book I, Chapter VII), Aristotle argues that the excellence for something is for it to perform its function (ergon) well. I take it that this is the argument to which you are referring. To summarize: The function argument states that everything, animate and inanimate, has a characteristic function. When something X has a function Y, then the function of a good X is to Y well.

For instance, the good of an axe is to chop wood well. Our organs also have characteristic function —  the good of our eyes is to see well, the good of our ears is to hear well, and so on. This can also be applied to tools, for instance. Consider a knife — the function of a knife is to cut and the excellence and good for a knife will be for it to cut well. The function of a bullock is to plough a field and so the excellence and the good for the bullock will consist in its ploughing the field well. 

The ultimate argument, of course, is that human beings as a whole have a function and in order to be good human beings, they must perform that characteristic function well.