The gens Julia (for which the proper plural masculine nominative is Julii, not Juli--Juli is the vocative singular), while being one of the oldest senatorial families, I don't think had any particularly good claim to being one of the "great families" of Rome. Caesar's branch of the family certainly was not particularly prominent, only appearing in the list of magistrates for the first time in 208, when the praetor Sextus Julius Caesar was sent to Sicily. The gens Julia was one of the older patrician families, but it was not one of the originals, those descended from Romulus' patres. The Julians were Alban nobility, who moved to Rome after Tullus destroyed Alba Longa. The gens Julia was therefore somewhere inbetween in status--they were not quite one of the patres, but not enrolled into the patrician order much later like the Sabine Claudii. The early Republic is largely without important Julians. The first Julian to hold the consulship was in 489, which was fairly early, although not in the first wave of consulships--this probably was not that much of a big deal, since it meant that the gens Julia was considered one of the nobiles, those families with consular ancestors, before most. The early Julians were rather politically active, holding a consulship about once a generation, but already by the end of the 4th Century this appears to have dropped off, and they largely disappear from our lists of magistrates. Most of those early magistracies are also not particularly eventful, and were monopolized by the Julii Iulli, the oldest branch of the family that seems to have gone extinct by the 3rd Century. Livy actually fails to mention the consuls of 489, when the first Julius Iullus held the consulship, and the last (and possibly most eventful, although there was a Julius Iullus who was part of the first decemvirate) Julius Iullus to hold office was dictator in 352, ostensibly to prosecute war against the Etruscans but really to appoint two patrician consuls in violation of the Lex Licinia, which codified in law what was already a tradition, to elect at least one plebeian consul each year.

Caesar's branch of the Julians does not seem to have been particularly remarkable at all. Their first consulship was not until 157, astoundingly late. The gens Julia as a whole sort of dropped off the map during the 3rd Century, holding only one consulship, that of Lucius Julius Libo in 267. That's somewhat important--everybody and his mother, at least among the original nobiles, could trace consulships back to the 5th and 4th Centuries, but the 3rd Century is when stuff actually starts getting almost contemporary. A total lack of consulships and only one recorded praetorship in the entire 3rd Century is pathetically low. Compare this to the gentes Cornelia, Fabia, and Valeria, who monopolized the consulship almost every year during the 3rd Century and into the 2nd. After 157 a Caesar didn't hold the consulship until 91, and then again in 90. Caesar's cousin Lucius held the consulship in 64, five years before Caesar's first consulship in 59. Caesar's own father never rose beyond the praetorship. Not a particularly impressive record, although a very old one--just about all the gens Julia had going for it before the start of its resurgence during the 1st Century was that it was really old, and lots of people could claim that

From what we can tell the members of the gens Julia mostly emphasized not their mortal ancestors but their divine ones. The New Pauly notes that "From at least the 2nd half of the 2nd cent. BC, the Caesares traced their lineage back to the goddess Venus via Iulus (= Ascanius) the son of Aeneas and their origin to his foundation Alba Longa; the dictator Caesar attached particular importance to this (Liv. 1,30,2; Serv. Aen. 1,267; Verg. Aen. 1,288; Dion. Hal. Ant. Rom. 3,29,7; Suet. Iul. 6,1; Tac. Ann. 11,24,2; coins: RRC 258 and 320; summarized: [4. 5-18])." Augustus emphasized it still further. This is very unlike, say, the gens Cornelia or the gens Claudia, which could point to prominent ancestors like Scipio or Appius Claudius. The gens Julia did have the distinction of being one of the "Trojan Families" from Alba, which gave them a lot of bragging rights as far as divine ancestry went. But not everybody believed in such things--Livy famously derides the Julian claim of descent from Aeneas. Now as for Caesar personally? The only actual ancestor, besides Venus and her progeny, that we know for a fact that Caesar honored openly was his uncle by marriage, Gaius Marius, which was a big deal since he publicly displayed images of Marius for the first time since Sulla's dictatorship