Yes, the character of the Prophet (peace and blessings of Allah be upon him) is the Qur'aan

Praise be to Allah.

Firstly: 

Yes, it is proven that 'Aa'ishah (may Allah be pleased with her) said that when describing the Prophet (peace and blessings of Allah be upon him). It is mentioned in the lengthy story about Sa'd ibn Hishaam ibn 'Aamir, when he came to Madeenah and went to 'Aa'ishah (may Allah be pleased with her) and asked her about some matters. He said: "I said: "O Mother of the believers, tell me about the character of the Messenger of Allah". She said: "Do you not read the Qur'aan?", I said: "Of course". She said: "The character of the Prophet of Allah  was the Qur'aan". I wanted to get up and not ask about anything else until I died" (Narrated by Muslim [746]).  

According to another report: 

I said: "O Mother of the believers, tell me about the character of the Messenger of Allah (peace and blessings of Allah be upon him)". She said: "O my son, do you not read the Qur'aan? Allah says:

{وَإِنَّكَ لَعَلَىٰ خُلُقٍ عَظِيمٍ}

'Translation' {And verily, you (O Muhammad are on an exalted (standard of) character} [al-Qalam:4]. The character of Muhammad was the Qur'aan" (Narrated by Abu Ya'la [8/275] with a saheeh isnaad). 

Al-Nawawi (may Allah have mercy on him) said in Sharh Muslim [3/268]: "What this means is that he acted in accordance with it, adhering to its limits, following its etiquette, paying heed to its lessons and parables, pondering its meanings and reciting it properly" (End quote). 

Ibn Rajab said in Jaami' al-'Uloom wa'l-Hukam [1/148]: "What this means is that he followed its etiquette and adopted its attitude. Whatever was praised in the Qur'aan, he was pleased with, and whatever was condemned in the Qur'aan he hated. It says in one report that she said: "His attitude was the Qur'aan, whatever it was pleased with he was pleased with and whatever it hated he hated"" (End quote).

Al-Munaawi said in Fayd al-Qadeer [5/170]:  "i.e., whatever the Qur'aan contained of commands, prohibitions, promises, warnings, and so on. 

Al-Qaadi said: "i.e., his attitude embodied everything that was contained in the Qur'aan. Whatever it regarded as good, praised or promoted was his attitude, and whatever it regarded as bad and prohibited, he would avoid". So the Qur'aan described his character" (End quote).

Secondly 

One of the rights that the Prophet (peace and blessings of Allah be upon him) has over us -especially in these days when his noble character is subjected to lies and distortions- is that we should mention some of his noble characteristics and praiseworthy qualities, so that the world might know that his is one of the greatest and purest of characters. 

Abu Haamid al-Ghazaali (may Allah have mercy on him) said in Ihya' 'Uloom al-Deen  [2/430-442]: "A number of the good qualities of his character which have been compiled by some of the scholars from the reports". Then he said: "He was the most forbearing of people, the most courageous of people, the most just of people, the most chaste of people.

His hand never touched the hand of any woman unless he owned her as a slave or was married to her or was closely related to her by blood (mahram).

He was the most generous of people, who never kept a dinar or a dirham with him overnight. If he had anything left over and he could not find someone to give it to before night came, he would not go home until he had donated it to someone who needed it.

He did not take anything from that which Allah had bestowed upon him except one year's supply of the simplest provisions, dates and barley, giving all of that for the sake of Allah. He was never asked for anything but he gave it, then he would go back to his annual supplies and donate from them to those who needed it more, then he might run out before the year ended.

He used to repair his own sandals and mend his own clothes, and he would help his family in the home and cut meat for them. He was the most modest of people and would not look anyone straight in the eye. He would respond to the invitations of slave and free alike, and accept a gift even if it was a cup of milk, and he would reward a person for it.

He did not eat food that had been given in charity, and he would respond to slave women and the poor when they asked him for something.

He got angry for the sake of his Lord but he did not get angry for his own sake. He would adhere to the truth even if that resulted in harm for himself or his companions. He found one of the best of his companions slain in an area where Jews lived, but he did not treat them harshly or do more than hat which is prescribed by sharee'ah. Rather he paid a diyah for him of one hundred camels even though some of his companions were in desperate need of just one camel. 

He would tie a rock to his stomach to ward off hunger pangs, and he did not refuse halaal food or and he would not eat reclining or at a table. He never ate his fill of bread for three days in a row until he met Allah, may He be exalted, as he would prefer to give away what he had rather than eat his fill, not because of poverty or miserliness. He would accept invitations to meals, visit the sick, and attend funerals.

He walked alone among his enemies without a guard. He was the most humble and quiet of people without being arrogant, the most eloquent without being long-winded, the most cheerful of countenance. He did not worry about worldly matters.

He wore whatever he found, and let his slave or others ride behind him on his mount. He rode whatever was available, sometimes a horse, sometimes a camel, sometimes a mule and sometimes a donkey. Sometimes he walked barefoot, with no cloak, turban or cap, visiting the sick in the furthest parts of Madeenah.

He loved perfume and hated foul smells. He would sit with the poor and offer food to and eat with the needy, honouring the virtuous and softening the hearts of people of status by treating them kindly. He upheld ties of kinship without favouring his relatives over those who were better than them, and he did not treat anyone harshly.

He accepted the excuses of those who apologized to him; he would joke but he only spoke the truth, and he would smile without laughing out loud. If he saw permissible play he did not denounce it, and he raced with his wife. When voices were raised against him, he bore that with patience.

He had slaves, male and female, but he did not eat or dress any better than they did. He did not waste time without striving for the sake of Allah or doing that which was essential to better himself. He did not look down on any poor person because of his poverty or chronic sickness, and he did not fear any king because of his power. He called both of them to Allah on equal terms. 

Al-Bakhtari said: "The Messenger of Allah (peace and blessings of Allah be upon him) did not revile any of the believers but he prayed that it might become an expiation and a mercy for them. He said: «I have been sent as a blessing, not a curse». 

If he was asked to pray against anyone, whether Muslim or kaafir, he would refrain from praying against him and pray for him instead. His hand never struck anyone. If he was given the choice between two things he would choose the easier option, unless it involved sin or the severing of family ties.

Allah described him in the Torah before He sent him, and said: Muhammad the Messenger of Allah, My chosen slave; he is not harsh or rough, and does not make noise in the marketplace. He did not repay evil in kind; rather he would pardon and forgive. Part of his attitude was that he would be the first to greet whomever he met, and if someone came to him with a need, he would be patient until the person was the first one to leave. If someone took him by the hand, he would not let go until the other person let go first.

In a gathering he could not be distinguished from his companions. Allah, may He be exalted, said: 

{فَبِمَا رَحْمَةٍ مِّنَ اللَّـهِ لِنتَ لَهُمْ ۖ وَلَوْ كُنتَ فَظًّا غَلِيظَ الْقَلْبِ لَانفَضُّوا مِنْ حَوْلِكَ}

'Translation' {And by the Mercy of Allah, you dealt with them gently. And had you been severe and harsh‑hearted, they would have broken away from about you} [Aal 'Imraan from the verse:159]. 

Allah bestowed upon him the best attitude and conduct, and the best way of dealing with people and situations, even though he was illiterate and could neither read nor write. He grew up poor in an ignorant desert land, tending sheep, an orphan with neither father nor mother. But Allah taught him all good characteristics and good ways, and taught him the stories of earlier and later generations, and that which brings success and salvation in the Hereafter and happiness in this world, and showed him the way to focus on one's duties and keep away from inessentials. May Allah help us to obey his commands and follow his example. Ameen O Lord of the Worlds" (End quote). 

No one should think that what we have mentioned above is no more than a nice story or an exaggeration that is not real, rather every single point mentioned is to be found in saheeh ahaadith that are narrated in the Musnads, Saheehs and Sunans, Whoever wants to know more may read al-Shamaa'il al-Muhammadiyyah by Imam al-Tirmidhi (available in English translation under the title (Shamaa-il Tirmidhi)). 

Fourthly: 

Finally, I urge you to seek help in your research by using modern computer programs, of which there are many, praise be to Allah. They will save you time and effort, helping you to find the hadith you want and learn its rulings. I also advise you to buy some comprehensive books which include the ahaadeeth of the Prophet (peace and blessings of Allah be upon him) and organize them by subject matter. Among the most important and accessible of them is Riyadh al-Saaliheen by Imam al-Nawawi and al-Targheeb wa'l-Tarheeb by Imam al-Mundhiri. The ahaadeeth are organized by subject, and compiled from all the books of hadith, and the scholars have commented on them and pointed out which reports are saheeh and which are da'eef (weak), such as Shaykh al-Albaani (may Allah have mercy on him). 

I ask Allah to reward you for your efforts and research, and I ask Him to help us and you to do that which is good. 

And Allah knows best.