I watched the show as well and since then have done more reading on the details of Octavian's life.

I think probably the biggest thing that stuck out to me in my sources was the image of him as a spirited and talented child who was often held back by his health but compensated for this with a skilled entourage. Portrayals of his coldness in the show are most likely drawn from his actions later in life.

Let's take a little closer look at the historical Octavian:

Octavian had a somewhat typical upbringing for someone of his political background. He went through much of the classical training of the elite and formed a close group of friends from amongst his political peers. The bonds formed between these individuals became very important later. The public also looked favorably on this band of companions as it harkened back to the famous group of friends that Scipio and other leaders had surrounded themselves with in the past. I mention this group here as it helped make up for some of Octavian's faults that you alluded to in your question. I would specifically call attention to Agrippa whose later generalship would win many victories in Octavian's name.

Now we can turn to the specifics about Octavian. We know that he was clearly smart and ambitious. He showed prowess in certain physical activities but more and more showed signs of some ill health. Octavian's mother was very protective of him for this reason. Nonetheless, he strived hard to excel.

Caesar, his great uncle, met Octavian in passing at first but would later spend more time with the youth. Caesar was a quick judge of a person's character and developed a fondness of Octavian, increasingly granting him favors.

Octavian went as far as traveling to Spain to follow Caesar on campaign but was delayed due to his health. Nonetheless he made the trip with his companions. Later Octavian was again brought low by his health while he was hosting games in Rome (perhaps due to heat stroke from being present all day). Caesar was very worried about the boy and inquired constantly about his recovery.

These spells of ill health would become a recurring feature in Octavian's life. After Caesar's death and in the ensuing civil war there would be several times when Octavian sought to lead forces but could not. I have not read what the official cause was but in my readings it basically seemed to happen almost like stage fright before an important battle. This isn't to distract too harshly from his character because in many instances Octavian strove to, and indeed did, act bravely.

It's during these weak moments that either careful planning or others would carry the day for Octavian. Without the help of Agrippa, it's hard to imagine things playing out in favor of Octavian the way they did. Once his rule was established, Octavian proved to be a very careful man. He was greatly concerned with the image of his rule and himself. In terms of politics, he was often ruthless and unforgiving, banishing or killing members of his family to make sure things went as he planned. I'm assuming it's this Octavian that they projected back into the show to make him seem cold hearted.

If you are interested in further reading, I would recommend the book Augustus: The Life of Rome's First Emperor by Anthony Everitt.