Was Rama aware that he is an avatar of Vishnu (during his lifetime) and acted as normal or he just tried to be good human being?
Yes, Lord Rama knew that he was an incarnation. There is an incident at the time of Sita Swayamvar where when he accidently broke Lord Shiva's bow. Lord Parshuram, who was also an incarnation of Lord Vishnu, came to know of this and came there started talking angrily about who has broken the bow.When Lord Rama came forward to calm him down, Parshurama asked Lord Rama to show who he actually is. And then Lord Rama showed him his Vishnu form and thus Parshurama calmed down and returned back. This incident is there in Valmiki's Ramayana.
There was also an incident in Uttara kanda of Ramayana described here, where Lord Yamraj came in disguise of a sage to tell Lord Rama that His time is over on earth and if He wants to stay more or end His life as human being. Following is the extract of what Yamaraj said to Lord Rama.
Thou didst use to protect the deities whenever they were assailed by any calamity. O lord of the universe, it is for that reason, on beholding the destruction of creatures thou wert born on earth to slay the Ten-necked demon. And at that time thou didst promise that thou wouldst live in the land of mortals for eleven thousand years. Thereupon thou didst assume a human form to carry out thine desire. Now that period is ripe and this is the proper time to inform thee of it. O great king, do thou wait in this land of mortals if dost thou wish to govern people for some time more. And if dost thou wish to repair to the region of immortals, do thou again lord over the deities in thy Vishnu form and let them be freed from anxiety.
Rama then agreed to depart the earth soon, for the welfare of the gods. So this clearly shows that Lord Rama knows about that He is God. Btw, this story is explained in detail in this answer.
Apart from this, there are certain other instances also which might have let him know that he is not a normal being, eg, He was appointed to free Ahalya from her curse, He was being worshipped (not only being helped) by Lord Hanuman, Vibhishana, etc.
Yes, Rama knew that he was an avatar of Lord Vishnu during his life time. It is mentioned in detail in the Yuddhakanda of Valmiki Ramayana. He was Lord Vishnu born as a human being. He incarnated to kill Ravana. As Ravana could be slain only by a human or a Vanara, other Gods didn't reveal his divinity even they know about this.
After completion of the war with Ravana and slaying him, Rama ignores Sita and asks her to leave him. Sita gives a taunting reply and decides to enter the fire in order to prove her fidelity.
Hearing the cries of these who were thus wailing, the virtuous minded Rama then became thoughtful for a while, afflicted as he was with melancholy, his eyes filled with tears. [6.117.1]
The bold part shows that even Rama was sad because Sita was entering into the funeral pyre.
Then the gods approach Lanka in their celestial cars to remind Rama his divinity.
Thereupon, Kubera the King of Yakshas, Yama the lord of death together with the deceased ancestors, Indra the lord of celestials Varuna the lord of waters, the illustrious Shiva the great deity who bears the device of a bull as his banner and having three eyes, Brahma the creator of all the worlds and the best among the knowers of sacred knowledge all these together reaching the City of Lanka in aerial cars, shining like the sun approached Rama.[6-117-2,3,4]
Lifting their long arms, their hands decked with ornaments, those excellent gods thereupon, spoke (as follows) to Rama who stood there, making a respectful salutation to them with his folded hands.[6-117-5]
They try to remind him his divinity. But Rama couldn't recollect this and considers himself as a human being. He asks them to remind what he forgot.
अब्रवित्त्रिदशश्रेष्ठान् रामो धर्मभृतां वरः |
आत्मानं मानुषं मन्ये रामं दशरथात्मजम् || 6-117-11
सोऽहं यस्य यतश्चाहं भगवंस्तद्ब्रवीतु मे |
Rama spoke as follows "I think of myself to be a human being, by name Rama, the son of Dasharatha. You, as a gracious Divinity, tell me that which I as such really am like this."
Then the creator of the world Brahma explains Rama about his real nature. He says in a verse:
अब्रवीच्छृणु मे वाक्यं सत्यं सत्यपराक्रम |
भवान्नारायणो देवः श्रीमांश्चक्रायुधः प्रभुः || 6-117-13
एकशृङ्गो वराहस्त्वं भूतभव्यसपत्नजित् |
Brahma said "Listen to my true word, O the truly brave lord! You are the Lord Narayana himself the glorious god, who wields the discus. You are the Divine Boar with a single tusk, the conqueror of your past and future enemies."
After this verse, Lord Brahma explains other avataras of Vishnu too. You can read them in the chapter 117 of Yuddhakanda
So, it is clear that Rama knew about his divinity during his lifetime. He knew it before returning to Ayodhya.
That's the only difference between him and us We are also a part of him but we don't realise it First chapter of Bhagwadmahapuran says that bhagwan Vishnu divided and started life on earth
In Tulsidas jis Ramcharitmanas Raam ji calls Agni devta and hands over Sita to him And Ravan abducted shadow of Sita
Supreme God Narayana “enacted” as Rama to honour Brahma’s boon to Ravana. When it comes to Rama read VALMIKI Ramayana (and nothing else)before giving your opinions. Lord Brahma prayed to Lord Rama in Yudda Kanda of VALMIKI Ramayana adressing Lord Rama as Lord Narayana
bhavaannaaraayaNo devaH shriimaaMshchakraayudhaH prabhuH || 6-117-13 ekashR^iN^go varaahastvaM bhuutabhavyasapatnajit | "You are the Lord Narayana himself the glorious god, who wields the discus. You are the Divine Boar with a single tusk, the conqueror of your past and future enemies."
THE INCONVENIENT TRUTH
That is NOT TRUE. Raam was unaware that he was an avatar of Vishnu.
Let me use The Valmiki Ramayan as my source. Unfortunately I cannot use the Hanumad Ramayan as it was destroyed by Hanuman as soon as he carved it on stone with his nails. Valmiki Ramayan is the earliest untainted source, and Valmiki, like Hanuman, has first-hand knowledge as he himself appears after his version of the Ramayan (Valmiki gave shelter to Mother Sita when she was cast out of the palace by Lord Rama, after the Ramayan).
Just before Mother Sita's Fire Sacrifice at the end of Valmiki Ramayan she says to Lord Rama:
"Aryaputra" and sobbed to him for comfort. Lord Rama's face showed a strange transformation of mind. None of those around him could understand him, not even Lakshmana and Lord Rama said to Mother Sita:
I have slain the enemy, I have recovered you. I have done my duty as a Kshatrya. My vow is now fulfilled... It gives me no joy to get you back for doubtfulness envelopes you like a dark cloud of smoke
Sita's eyes flashed fire and she shot back "Unworthy words have you spoken, my ears have heard them and my heart is broken..."
Mother Sita calls for the fire to be set and leaps into the fire screaming:
Ye Gods, I bow before you. Oh Rishis, I bow to you. Oh Agni, you atleast know my purity and will take me as your own
Agni the God of Fire rose in his own body, lifting Mother Sita in his arms with all her jewels and clothes intact, and presented her Rama.
Lord Rama said:
"Who am I? All that I know and can tell you is that I am Rama, son of Dasaratha. You know who I am and whence I came. It is you who must inform me"
Glory Glory Glory to the Divine Mother in whose form all women are created.
SOURCE: VALMIKI RAMAYANA RETOLD BY C RAJAGOPALACHARI - 1957, BHARATIYA VIDHYA BHAVAN