Like Lord Krishna in Mahabharata (geeta) made clear about his divinity, did Rama also show his divine aspect or claim that he was an avatara or god ? Did he make any claims or statements that he was enlightened or an avatara?
No, Rāma did not make any such statements. In fact, he said quite the opposite:
आत्मानं मानुषं मन्ये रामं दशरथात्मजम् || ६-११७-११
सोऽहं यस्य यतश्चाहं भगवंस्तद्ब्रवीतु मे |
ātmānaṃ mānuṣaṃ manye rāmaṃ daśarathātmajam || 6-117-11
so.ahaṃ yasya yataścāhaṃ bhagavaṃstadbravītu me |
I think of myself to be a human being, by name Rama, the son of Dasaratha. You, as a gracious Divinity, tell me that which I as such really am like this.