This is a great question. There are many deeds that Aeneas has done and obstacles that he has overcome to be the hero of the epic poem. In light of this, let me name a few of them.
First, in the burning of Troy, he shows that he is dutiful. He takes care of family and the household gods and he flees. He has his father Anchises on his back and his son and wife in tow. His wife dies along the way. This is found in book 2.
Second, we see that one of the easiest things for Aeneas to do was to settle down in Carthage with the beautiful Queen, Dido. He is tempted. However, he knows that he has been destined by the gods to establish a new land. So, with a sense of duty, he leaves. We read of this in book 4.
Third, Aeneas and Turnus have a final bout to the death and Aeneas bravely fights for this people and the hand of Lavinia. Aeneas comes out victorious. We read of this epic battle in the final book.