Well, it would depend on who you asked. I'm not particularly familiar with Anglo-Saxon England, but a good place to start would be with the Church and, especially after the reforms of the eleventh century, the Papacy (as the Church itself was not monolithic and therefore cannot be relied on to issue one overarching thesis of what constitutes a "good" king). Take for example Pope Gregory VII's (1073-1085) views on kingship, which we can extrapolate from his vast correspondence with a wide range of royal dignitaries throughout Europe, from the French monarch Philip I and the German anti-king Rudolf of Rheinfelden to Prince Isjaslav of Russia and King Toirdhealbhach of Munster. His views regarding the institution of monarchy were complex, at once dynamic and rooted in ancient tradition: while he would frequently allow for compromise, reconciliation, and innovation in his dealings with kings, Gregory was at the same time steadfast in portraying his role as analogous to that of an Old Testament prophet, with the power to judge errant Christian kings. To do so, he needed to define the characteristics of a "good" king.

Several of Gregory’s letters presented charity as one of the principal attributes of a good Christian, as in a 1074 message to his allies the Countesses of Tuscany, in which he asserted that

In truth, from love of God and by holding dear one’s neighbour to help the wretched and assist the oppressed – this is something that I place before prayers, fasts, vigils… for I do not hesitate with the Apostle to set true charity before all virtues.

This clearly indicates the importance Gregory attached to charity. With regard to kings, Gregory declared that he would rather die than respect “ungodly” magnates over the charitable, suggesting that he saw charity as a crucial trait of a proper Christian king.

Another virtue that Gregory regarded as a vital trait of a suitable king was obedience; indeed, this value is mentioned more than any other in his letters. Gregory believed that kings had a duty of obedience to St Peter and his worldly vicar, as evidenced in a letter sent to King Henry IV of Germany in 1075, shortly before the breach between king and pope, in which Gregory made his apostolic blessing conditional upon Henry showing obedience to Petrine authority “as befits a Christian king.” When Henry submitted to Gregory at Canossa in 1077, the pope rejoiced at his willingness to “walk in the ways which he has promised us.” Moreover, after renewed hostility between Henry and Gregory, the record of the Lent synod of 1080 acknowledged Henry’s rival Rudolf of Rheinfelden as king because he was “ready to be obedient to me [Gregory] in all things.” In some cases, obedience could be demonstrated by military service in aid of the church, as evinced by Henry’s exhortation to the Danish king to combat “the heathen and the enemies of God.”

Gregory also regarded humility as a prerequisite for kingship, and castigated monarchs whom he saw as prideful. Gregory used the likes of the Israelite King David and the Roman Emperor Constantine the Great as ancient paradigms of this regal quality, though he also praised contemporary kings like Alphonso VI of León-Castile for their “exalted humility.” The pope was much impressed by Henry’s show of humility at Canossa in 1077 when he “laid aside all kingly adornment,” deeming it an essential precondition for his restoration to communion and to the royal dignity.

Gregory’s correspondence also enshrined righteousness as a fundamental moral attribute of a true king, linking it with both religion and justice and asserting that it depended on pontifical direction and observance of the law. A number of letters to royal recipients illustrate Gregory’s understanding of the concept in relation to kingship, including one in which he called upon the kings of Spain to “perform righteousness and to protect the liberty of the Christian faith.” Thus he was greatly concerned with the demonstration of proper morality in Christian kingdoms. Gregory moderated this advocacy for iustitia by calling on kings to be merciful in certain situations. Gregory’s correspondence with King Harold Hein of Denmark twice referenced righteousness in conjunction with mercy, in one case calling upon the king to treat women accused of witchcraft with restraint. Therefore, it is evident that Gregory believed that kings should dispense justice with respect for religion but tempered by compassion.

In addition to exhibiting proper moral suitability for the office, Gregory also expected kings to perform certain duties and thereby prove to be serviceable. Many of his letters reveal a fear that kings were failing to properly protect the church, and in some cases openly attacking it. To remedy this, Gregory’s letters detailed a number of ways in which royal obligations to the church can be fulfilled, including safeguarding the election of bishops. In a letter to the king of England, Gregory sought the king’s assistance in electing a successor to the ailing archbishop of Rouen, while a later letter to the king of France promised “apostolic good will, as becomes a Christian king... if you shall reverently show yourself diligent and devout in matters of ecclesiastical business” by expelling the deposed archbishop of Rheims and protecting the electors of his successor.

Another obligation of an ideal Christian king as envisaged by Gregory was the defence and augmentation of church properties. Several letters alluded to this royal responsibility, such as a pastoral to the king of Denmark, which implored him to “protect, foster, and keep [the church], and to seize her from the jaws of wolves that lie in wait.” Another example is a letter sent in response to Henry’s expression of penitence, in which Gregory reminded the king that “you hold royal power duly, if you bow down the height of your dominion to Christ the king of kings to bring about the restoration and the defence of his own churches.” Further examples sent to monarchs in all areas of Christendom – France, Hungary, Sweden – throughout Gregory’s pontificate testify to the significance this form of service had in the mind of the pope.

Gregory also judged that kings should heed the advice of good counsellors if they were to govern in a serviceable manner. His letter to King Philip I of France exhorted him to “despise the counsels of the wicked and above all to abhor any association with excommunicates.” For this reason, Henry’s refusal to dissociate himself from the five counsellors excommunicated by Pope Alexander II in 1073 for simony was an extremely contentious issue in the early years of Gregory’s papacy, and one which he complained about frequently in his correspondence. When Henry’s penitential letter offered a chance of reconciliation, the pope took the opportunity to instruct the king to “take for yourself such counsellors as may not love your possessions but yourself, and who may consult your salvation and not their own gain.”

Gregory placed an especial value on a king’s responsibility of maintaining peace in his kingdom, and his letters attest to this preoccupation. An example of this is his view on the efforts of the brothers of King Harold Hein of Denmark to convince the monarch to commit part of the kingdom to them. In a letter to the Norwegian king Olaf III, Gregory warned him not to assist the brothers in doing so, as such a division would lead to internal strife and consequently lead to “the harm of the kingdom... the disorder of the Christian people... the destruction of churches... the desolation of the whole land.” Thus it is apparent that Gregory championed the stability and continuity of established regimes in order to prevent the internecine warfare which would inevitably result from dynastic rivalries.

A corollary of Gregory’s assessment of the importance of virtues such as charity was that he thought that the care of the poor and the clergy should be a priority among the duties of an ideal king. This was a recurrent theme in his correspondence, considering that it warranted mentioning in letters to royal figures ranging from the kings of Denmark, Norway, and France to the Queen of England, among others. The Swedish kings Inge I and Halstan were instructed to bestow “compassion upon the poor and afflicted,” while Queen Adelaide of Hungary was called upon to protect “the poor, orphans, widows, and all who are unjustly oppressed.” The fact that this obligation was mentioned so often signifies its importance to Gregory. Failure to carry out these duties would make a king liable to be deposed, as in the case of Childeric IV, who, according to Gregory, was dethroned “not so much for his iniquities as for the reason that he was not useful for so great a power.”

To Gregory’s dismay, exemplars of his ideal king – a monarch who would be morally suitable and capable of fulfilling his obligations to the apostolic see – were to be found mostly in the past, and therefore he was unwavering in exhorting his contemporaries to imitate the pattern of their royal antecedents.

Approaching the character limit so will include sources in a separate comment.

When you say you are "reading Beowulf" what do you mean? Framing the retelling by specifically citing a translation or a reproduction that you are basing this off of might be very helpful. Beowulf, as you well know, is an extremely old tale. Who transcribed/translated it and when will provide a great deal of context towards answering this question.

It is very likely that you are reading a modern take on an ancient story, and analyzing the mindset of the author is important to your question, and therefore is important to the answer.

Personally, I feel you are asking something far too subjective to answer in the context of this subreddit, and that you might have better luck on an anthropology or sociology board. But, it's worth looking into!

EDIT: Or you can rephrase to ask in a tighter light