In the earlier days of the Republic, senate membership came with election to a "curule" magistracy, traditionally curule aedile, praetor, consul, magister equitum, and dictator, although there is a lot of disagreement about how exactly the senate was filled in those times. By the time of Caesar, the senate had both greatly expanded, and membership requirements had been pushed. Tribunes of the Plebs were added as automatic qualifiers some time in the last decades of the 2nd century BCE (I'm not sure when, honestly). In 81 BCE, Sulla made all elected quaestors automatic senators, and so we find that Cicero, a novus homo, was in the senate after his quaestorship in 75 BCE. This was part of a general expansion of the senate machinery under Sulla, necessary to replenish the ranks after a long period of depletion due to foreign wars, civil wars, and proscription.

Even after Sulla, it seems that the two Censors had final say over the senatorial roll, and that those who had automatically qualified by their magistracies were permitted to attend and to speak, but were not "true" senators until they had been accepted by the censorial process (See Lintott, Constitution of the Roman Republic 1999, 71ff). It is also possible (probable) that by tradition, and even into the Late Republic, the censors could and did admit members of the elite families, the nobiles, who were not old enough to be elected to positions, or had for whatever reason not chosen to run. Largely, the men who could possibly be elected to the quaestorship were from this same pool--there was no possibility of a homeless pleb "pulling himself up by the bootstraps" and getting elected quaestor. So yes, there were certainly men in the ranks at various points who were there based on their family status, but on the whole, the ranks were made up of current and former magistrates. A fellow who had become curule aedile could continue to sit in the senate for decades afterwards, even if he did not manage to move further up the ladder to the praetorship.

There are really no solid mentions of property qualifications for membership before the Principate. This means that in theory, Caesar could have been in the senate even as a young man. He was of course from the famous gens Iulia, of ancient patrician pedigree, his father had achieved relative success, and his mother was from the powerful and prestigious gens Aurelia. Still, his family branch was nothing particularly special in the first half of the 1st century BCE. Even more relevant, his position was incredibly precarious while Sulla was alive: he had been proscribed for refusing to divorce Cinna's daughter Cornelia. He only returned to Rome upon the news of Sulla's death, and even then he stayed out of politics and focused on his oratory in the courts. We hear nothing of any sort of activity in the senate. Thus most assume, probably rightly, that he entered the senate upon his election to the quaestorship in the middle of 70 BCE, to be served in the year 69 in Hispania Ulterior under the governor Antistius Vetus.