Basically, Greek classical convention didn't employ a surname, appending a reference to the city of origin or dynastic number if context required it. The Romans, and other Latin tribes, placed a great value on tracking patrilineal legacy once Rome and other city states became prominent. In Roman usage, the use of these surnames, as we would term them today, originally indicated the patrilineal descent from one of the particular tribes of Rome, though as Rome expanded their territory and citizenship outside the Latin areas, many of them 'adopted' traditional surnames, eventually making that distinction unclear. Still, the presence of a surname (and middle name, for that matter), in Rome continued to describe social and familial status, and eventually first names became of secondary importance, often being reduced to a small pool of traditional names, often with male children simply taking the father's name and so on for generations, with females even reduced to names indicating order of birth. Thus, the second and last name in conjunction with each other would tell you the status and origin or a particular individual, and differentiate them from the many, many other unrelated individuals who might share a common first name. For example, in relation to your example, that individual was actually named Gaius Julius Caesar, but just like in his day, we immediately know which Caesar you mean just by using Julius, even though Caesar eventually became more of a title than a name. So, to sum up, in short, Greeks didn't use surnames in that fashion as a cultural difference from Romans, who spread the practice through the Republic and Imperial eras. There's actually record of early Greek-Roman contact that can be interpreted as the Greeks being confused by the naming convention.
Source:
Benet Salway, "What's in a Name? A Survey of Roman Onomastic Practice from c. 700 B.C. to A.D. 700", in Journal of Roman Studies, vol. 84, pp. 124–145 (1994).