Count Baldassare Castiglione's The Book of the Courtier (1528), as your question suggests, is a detailed discussion among a group of Italian aristocrats, ladies and gentlemen, about the physical and mental qualities of an ideal courtier in sixteenth century Italy, one who is both an ornament to his prince and a counselor.
If we were to encapsulate the qualities of the ideal courtier, we would describe him as something close to a "Renaissance Man," someone who is skilled at all physical activities, especially those dealing with military life, and, equally important, writing, speaking, music (singing and playing multiple instruments), swimming, dancing, fencing, horsemanship. In addition, he must be morally perfect, modest but brilliant, a font of wisdom for his prince, and he must be able to do everything brilliantly without appearing to have expended any effort—in Italian, the word is sprezzatura, the art of doing everything without trying hard.
The basic quality of the ideal courtier, one that causes a lot of complex discussion among the aristocratic group, is, not surprisingly, that he be of noble birth:
I wish, then, that this Courtier of ours should be of noble birth and of gentle race; because it is far less unseemly for one of ignoble birth to fail in worthy deeds. ... (I:14)
In short, a common person would not be embarrassed to fail, but a noble person would be horrified. We need to keep in mind that, because the people "creating" this ideal courtier are all nobles, they would naturally choose from their own class.
A second, but very important, characteristic is that the Courtier be a warrior:
... the principal and true profession of the Courtier ought to be that of arms; which I would have him follow actively above all else, and be known among others as bold and strong, and loyal to whomsoever he serves. (I:14)
In Italy in the sixteenth century, it was common for armies to be made up of mercenaries rather than citizen-soldiers, so the concept of loyalty "to whomsoever he serves" is important because the warrior is fighting for money, not ideals or conviction, and his loyalty is tied directly to his wallet.
But this courtier-warrior cannot just be a soldier, he must show moderation in all other pursuits:
Therefore let the man we are seeking, be very bold, stern, and always among the first, where the enemy are to be seen; and in every other place, gentle, modest, reserved, above all things avoiding ostentation and that impudent self-praise by which men ever excite hatred and disgust in all who hear them. (I:16)
A refrain that goes throughout the work is the emphasis on modesty in all things, and this modesty carries over to eating and drinking modestly so that the courtier is never accused of any kind of excess. His skills in all things physical and mental should be perfect, but he can never be seen or heard to express his superiority. Again, the concept of sprezzatura applies: perfect execution without apparent effort.
Throughout Book II, the aristocrats discuss the Courtier's language and writing ability at great length, going so far as to discuss the elements of Tuscan Italian that make it either a good or bad choice for the Courtier's dialect, and even discussing borrowing vocabulary from other languages:
In such fashion I would have our Courtier speak and write; and not only choose rich and elegant words from every part of Italy, but I should even praise him for sometimes using some of those French and Spanish terms that are already accepted by our custom. (II:32)
This is actually more remarkable than it may seem. In sixteenth-century Italy, and especially in Tuscany where this conversation occurs, various armies from France and Spain are engaged in "political" struggles in Italy, and for a Tuscan aristocrat to encourage the use of French and Spanish is, at the least, fair-minded and perhaps politically useful.
In Book IV, which is sometimes considered "tacked on" to the first three books as an afterthought—if so, it is a good one—Castiglione adds an important characteristic of the ideal Courtier, who will be able to guide his prince to goodness
... and thus be able always to disclose the truth about all things with ease; and also to instill goodness into his prince's mind little by little, and to teach continence, fortitude, justice, temperance. ... (IV:250)
In short, this ideal Courtier—in addition to being a perfect musician, rider, dancer, and warrior—can lead his prince to become, well, the ideal Courtier but with much more power. He can, if he moves quietly, help create the ideal ruler in the prince he serves, a far cry from the job description established in the early books.