The Quran describes the characteristics of a 'perfect man' in many verses, such as verse 48 of chapter al-Fath,
You see them bowing and prostrating [in worship], seeking Allah’s grace, and [His] pleasure. Their mark is [visible] on their faces, from the trace of prostration.1
In other places the Quran says,
[The faithful are] penitent, devout, celebrators of Allah’s praise, travelers [for His cause], who bow [and] prostrate [in prayer], bid what is right and forbid what is wrong, and keep Allah’s bounds—and give good news to the faithful.2
[They are] patient and truthful, obedient and charitable, and they plead for [Allah’s] forgiveness at dawns.3
In Surah al-Furqan, a detailed and comprehensive discussion about the attributes of Allah’s distinguished servants is brought up under the title of ‘The servants of the Beneficent (Allah)’,
The servants of the All-beneficent are those who walk humbly on the earth, and when the ignorant address them, say, "Peace!" Those who spend the night for their Lord, prostrating and standing [in worship]. Those who say, "Our Lord! Turn away from us the punishment of hell. Indeed its punishment is enduring. Indeed it is an evil station and abode." Those who are neither wasteful nor tightfisted when spending, but balanced between these [two extremes]. Those who do not invoke another deity besides Allah, and do not kill a soul [whose life] Allah has made inviolable, except with due cause, and do not commit fornication... Those who do not give false testimony, and when they come upon frivolity, they pass by with dignity. Those who, when reminded of the signs of their Lord, do not turn a deaf ear and a blind eye to them. And those who say, "Our Lord! Give us joy and comfort in our spouses and offspring, and make us imams of the God-wary." Those shall be rewarded with sublime abodes for their patience and steadfastness, and they shall be met there with greetings and ‘Peace,’4
We can conclude from the verses cited above that one who has attained the state of perfection observes both his individual and social duties, maintains a good relationship with both his Creator and people, and acts upon religious obligations and refrains from unlawful things. A perfect man is someone who brings his talents and abilities from potentiality into actuality, and tries to fortify and improve them equally, taking into consideration the necessity of maintaining a balance between his talents and abilities, for one who fails to do so cannot be called a perfect man however righteous and good he might be.
References:
1. The Holy Qur'an, 48:29.
2. The Holy Qur'an, 9:112.
3. The Holy Qur'an, 3:17.
4. The Holy Qur'an, 25:63-75.