Having just finished Julius Caesar's biography and doing some additional research, I found few references in regards to Caesar's relationship with his grand-nephew Octavian.
Considering he adopted Octavian and named him heir, I assume they at least must have met on numerous occasions. However, the apparent lack of historical material also leads me to believe that it was simply the best option Caesar had in dynastic succession.
Hi R_O,
The most important thing to know initially about Caesar and Octavian (since we're discussing events prior to his imperial ascension, I'll be referring to him as Octavian) is that Octavian was Caesar's great-nephew. His mother, Atia, was Caesar's niece, but Octavian didn't formally meet the man until he was in his late teens. Caesar took a quick liking to the boy, even granting Octavian's friend and (future general) Agrippa's brother a second clemency on his nephew's request. When Caesar left to fight a war in North Africa around the year 46 BCE, Octavian had planned to accompany him, but fell ill and his mother forced him to stay behind. Upon his recovery and his mother's blessing, the young man raced to meet his uncle and assist him, but found that the war had already been won by Rome. Despite this, Caesar was incredibly impressed by Octavian's courage, as most aristocratic men began their journey to the front lines only after they knew the war had been won. Octavian came even though he had no idea how the war was going. Caesar was so pleased that he asked Octavian to ride with him and Marc Antony back to Rome.
After the war in North Africa, Augustus soon accompanied Caesar to Spain in 45 BCE to fight the Pompeian forces, where Caesar was again impressed by his competence. Caesar took it upon himself to bolster the young man's career upon their return. He saw the young Octavian's incredible intelligence, ambition, and cunning, all making up for his physical weakness and frequent illnesses. After Caesar's murder, many assumed Marc Antony would be named his heir, but Caesar had been disillusioned with Antony's ruling ability after his terrible leadership of Rome during his term as Caesar's magister equitum (master of the horse/dictator lieutenant). At the reading of Caesar's will, it was discovered that the late dictator had named Octavian his legal heir, and even adopted him posthumously. At this point, Octavian took the name Gaius Julius Caesar Octavianus to cement his relation to his adopted father, and took up the mantle that Caesar had left.
Now, what does this tell us? I would argue that given the historical information here, it makes logical sense that the relationship resulted in the elevation of Octavian. Here we see a growing familial relationship, that while only captured in snapshots, paints a picture a camaraderie of considerable depth. This cultivation of family ties between the two men is interesting in its genuine nature. Octavian had no reason to believe that Caesar would choose him as an heir, and so this relationship grew organically. There seems to have been real love (or at least happy familiarity) between Octavian and Augustus as an uncle and nephew.
Sources:
Firth, John Benjamin. Augustus Caesar and the Organization of the Empire of Rome. GP Putnam's sons, 1903.