If you want really "primary" primary sources, that's difficult. You don't have much written for maybe a century after the fact. There might be something in Paterculus (writing under Tiberius), and then you have Appian writing in the 2nd century CE, Plutarch's biography of Antony, also 2nd century, and then all the way down in the 4th century, Cassius Dio.

My guess is Plutarch is probably the most important source. While I'm not a Shakespeare scholar, I understand Shakespeare's Roman plays drew largely from Plutarch, which had been recently translated into French.

If you want contemporaneous sources, you'll be looking for inscriptions, coins, and stuff like that. Those are technical fields that take some study to understand. In that case, I would suggest a secondary scholar: Zanker, The Power of Images in the Age of Augustus (Review here). Zanker looks at what we have left of the propaganda both Octavian and Antony produced in the period leading up to the war at Actium. It's not the dryest academic book, but it might take a little work for someone unfamiliar with coinage, frescos, and other kinds of ancient images.

I will suggest you look at Shakespeare as well. Again, I'm not a Shakespeare scholar, but I have no doubt Shakespeare was the source material for the movie, and you may benefit from doing some reading on Shakespeare's sources (again, I bet money on Plutarch).