Here is a mosaic created by the Romans in honour of Alexander the Great, built in 99bc. The Romans were fascinated by the powerful and successful general, and there are various quotes of famous Roman generals, referencing Alexander's courageous acts. This mosaic is a visual representation of the Battle of Issus, the first battle for his conquest of Persia. You can see Alexander the Great on the left, looking the enemy in his eye whilst horseback. On the right we see the Persians, under the rule of Darius III of Achaemenid Empire. He wears a traditional headdress and holds a dory (Persian spear). We also see Alexander striding in to battle with courage and without mercy, guiding his fellow comrades into a war that Alexander told his soldiers with body language to fight with bravery. A quote from a Roman General states, "For a few days, we were encamped in exactly the same place that Alexander occupied when he was fighting Darius at Issus..." This shows that Alexander was famous amongst the Roman community, and his legacy passed down through the years.
Here is an astronomical diary mentioning Alexander the Great's triumphant entry to Babylon with his army, as well as stating the defeat of Darius III of Persia. With the inscriptions mostly stating what the Athenian orator observed in the sky, it also mentions how Alexander the Great had defeated Darius III of the Achaemenid Empire. The diary reads, "On the 24th [1 October], in the morning, the king of the world [Alexander] erected his standard and attacked. Opposite each other they fought and a heavy defeat of the troops of the king [Darius] he [Alexander] inflicted. The king [Darius], his troops deserted him and to their cities they went. They fled to the east." Followed by, "The significance is: The son of the king will become purified for the throne but will not take the throne. An intruder will come with the princes of the west; for eight years he will exercise kingship; he will conquer the enemy army; there will be abundance and riches on his path; he will continually pursue his enemies; and his luck will not run out." This refers to Alexander the Great being described as powerful, and the thought of him winning control over any opposition in which challenged him.
Here is a video stating whether or not Alexander the Great was a "great" man. He brings up how he was supposedly a son Zeus, the leader amongst the gods, and how he watched over him, guiding him through greatness, conquering whatever he wanted to gain power. Across 33 years, Alexander had conquered all of Greece and the Asia Minor. But maybe he earned the name "great" from his followers, such as the Romans which came later, but deeply admired his work. It seems highly unlikely that he would name himself that, it would only seem logical if he gained it from his followers. You have to be extremely over-confident to be called Great, you would have to defend your name from nations whom are willing to hold the "great" title. I believe that Alexander the Great got his name by conquering all of Asia and Greece, but I believe that the Roman General in which followed his actions had named him that, not himself.
Another historians account on whether he was "great" or not is James Romm. he states, "There are only a few people in history who are universally known as "the Great." This is followed by him stating, "The word "great" in this context, to my mind, is always positive -- implying both that Alexander's achievements were huge in scale, and that his nature was heroic and awe-inspiring." He is saying that only a few people in the Ancient who were called "great" but, is it truly a heroic name. Most people who were named "great" weren't actually so great. They caused chaos and trauma wherever they set foot, and in Alexander time, he did a lot of that. He used to burn down buildings, as well as knock them down. Him and his army killed many innocent people, but also did bring strong warriors into their army, to fight them.