"Alexander the great" was known as a man who would conquer a city then move on to the next. He would leave his generals behind to rule the city. True to his nature in this context means he would always leave a city before he witnessed any buildings being built. This was in his nature (he always did). I hope this is what you wanted to know :)

it is kinda repeating, true to some ones nature is what is natural for a person. being yourself is being true to your nature, doing what you do, best is another way of saying the same thing, they are saying he is being him self.