I took the chance to make a little summary of ancient sources on Caesarion.
• Cicero in his letter to Attico (10 May 44 BC) writes “I am sorry for dear Tertia's miscarriage: we want as many Cassii produced as Bruti. I wish it may be true about the Queen and that Caesar of hers”.
• Nicolaus of Damascus (contemporary of Augusts) in Life of Augustus 20 writes “Furthermore, all sorts of rumors were being bandied about in the crowd, some telling one story, others another. Some said that he had decided to establish a capital of the whole empire in Egypt, and that Queen Cleopatra had lain with him and borne him a son, named Cyrus, there. This he himself refuted in his will as false. Others said that he was going to do the same thing at Troy, on account of his ancient connection with the Trojan race.”
• Suetonius (≈ 70-126 AD) in Caesar 52 says that Caesar “allowed her to give his name to the child which she bore”.
• Plutarch (≈ 46 – 127 AD) in Caesar, 49,10 says that Caesar “leaving Cleopatra on the throne of Egypt (a little later she had a son by him whom the Alexandrians called Caesarion), he set out for Syria” , and in Antony 54,4 says that “Caesarion was believed to be a son of the former Caesar, by whom Cleopatra was left pregnant”.
• Cassius Dio (≈ 165-229 AD) in Book 47, 31,5 says “Cleopatra also, on account of the aid she had sent to Dolabella, was granted the right to have her son called king of Egypt; this son, whom she named Ptolemy, she pretended was her son by Caesar, and she was therefore wont to call him Caesarion”.

Then we are told that Antony always maintened that Caesarion WAS Caesar’s son.
We are also told that G. Oppius produced a pamphlet where he proved that Caesar WAS NOT the father of Caesarion.
Finally, we know that Augustus killed the poor boy, while he let Antony’s children live.

I think it’s understandable the position of Antony: he wanted to be the practical heir of Caesar (in terms of power and control over the empire) so to have in his family the son of Caesar would give him a stronger stand and a legal claim on Caesar’s legacy.
Augustus, whom I don’t know if he expressed his idea on Caesarion, did not want to run the risk of the boy claiming his right to rule on Rome, once he grew up. So… just to avoid possible problems, it was easier to get rid of Ceasarion indipendently from who his father was.

What puzzles me is Gaius Oppius. Being an intimate friend of Caesar he must have known what he was talking about. But again, Caesar was not there any more, so he couldn’t dispute what Oppius said. But why should Oppius produced a pamphlet denying the paternity of Caesar? On whose side was he, Augustus or Antony or none?