I should really go through all the threads where I talk about patricians and plebeians one of these days and add them to the FAQ

The patricians as a social order were not a "class" in either the political or economic sense, not as we'd understand a class to be. The patricians were all families that traced descent from one of the patres, the original couple hundred senators under Romulus. A couple generations later the Alban nobility and some Etruscan families were added to the order--this is how the gens Julia entered, as it was one of the "Trojan families" that had been enrolled after the destruction of Alba Longa. A little later some of the other Italian nobility was added to the order, particularly a number of Sabine families, including the gens Claudia. Descent from one of these ancestors was the only requirement for admission into the patrician order. By the Twelve Tables, patrician status could only be transferred to children who could claim both of their parents as members of the order--because of this, the patrician order had enormous difficulty maintaining itself. Already by the Punic Wars many of the minor patrician families seem to have died out, and by Caesar's day only fourteen patrician gentes still had existing patrician lines out of over fifty known in earlier periods, although many of these families still had existing plebeian lines.

Initially patricians held certain political privileges, but these were removed from them almost immediately. Tradition held that originally only patricians could hold the consulship, but this is certainly false--the Fasti show several plebeians as consuls, including Brutus himself (the gens Junia was plebeian). The Conflict of the Orders was largely resolved by the end of the 4th Century, at which point the patrician monopoly on magistracies had been eradicated--by the end of the Conflict of the Orders the only magistracies exclusive to the patricians that had no plebeian analogs (such as the aedileship, for which there were plebeian and curile aediles) were certain priesthoods. By Caesar's day the distinction was politically meaningless--plebeians could hold every senatorial magistracy and several, such as the incredibly powerful Tribune of the Plebs, could only be held by a plebeian. Although patrician families continued to hold an inordinate amount of influence during the 3rd Century, despite the loss of their political exclusivity, by the 2nd Century that had already largely eroded away and by the 1st Century it was completely moot. Besides magisterial privileges, the Conflict of the Orders had definitively resulted in the massive increase of power among the comitia, the citizen assemblies. The comitia elected all the magistrates, who were the only ones who had official political power (the senate, though influential, had no official legal powers and was an advisory body composed of former magistrates), and all citizens could (and were expected to) vote in the comitia--indeed, certain comitia only allowed plebeians to vote. The comitia centuriata, the most important of the assemblies, was initially heavily weighted towards the patrician order even after their loss of political privilege, since it was divided up according to property classes, but while the sex suffragia, the highest centuries within the comitia centuriata always held enormous power, we will see that wealth was hardly restricted to the patricians, and already by the Punic Wars an enormous amount of the centuries in the first property class are composed of plebeians--by Caesar's day the first property class was composed almost entirely of plebeians.

As said above, patricians did not have a monopoly of wealth by any means. Indeed, several patricians are known to have been severely impoverished, most famously Sulla and Catiline (and possibly Caesar as well). Eligibility for senatorial magistracy required 200,000 sesterces under Augustus, and we should expect a similar amount going back quite far. This was a decent amount of money, but hardly a fortune--Cicero's house on the Palatine, for example, cost him 3.5 million sesterces, and while it was his pride and joy it wasn't exactly the most expensive house in the city (Sulla's daughter sold a villa for 10 million sesterces!). An entire sub-class of the plebeian order, the equites, was composed of those plebeians who met the financial requirements of the senatorial order but had no senatorial ancestors and had no particular ambitions towards senatorial magistracy. Many of these equites, particularly during the Principate, became fabulously wealthy--Cicero's friend Atticus, for example, was stupidly rich. And plenty of plebeians within the senatorial class were incredibly rich--Crassus and Pompey were both plebeians.

This is not to say that there were not class distinctions in the city. There most certainly were, but what mattered was not whether you were a patrician or a plebeian but whether you were a senatorial family or not. Enrollment into the senate was automatic for anyone who had been elected to a senatorial magistracy (quaestor and above)--once your year of office expired you were enrolled into the senate, the advisory body to the magistrates. The senatorial class was composed of both plebeians and patricians, since both were eligible to all senatorial magistracies. Indeed, the plebeians certainly formed the majority of the senate, given the decline of the patrician families and the tradition of having plebeians as well as patricians serve as colleagues. By Caesar's time, actually, it's hard to find patricians in the senate at all--the vast majority of the names you'll find in Cicero, Suetonius, Sallust, etc. are all plebeians. Both the Bruti, Crassus, Pompey, Cicero, Cassius, Antony, Lucullus, Cato, Hortensius, Lepidus--these are all plebeian names! Within the senatorial class there was a sub-class, the nobiles. These were families who not only had senatorial ancestors but consular ancestors as well--the senatorial class was always relatively exclusive, since it was naturally hard to get yourself elected politically (and harder still if for some reason the senatorial class got everybody to vote against you), but the nobiles were absurdly so, so much so that the phrase novus homo, "new man," was coined to describe those rare individuals like Cicero and Marius who obtained a consulship without any consular ancestors (technically a novus homo was someone who obtained a senatorial magistracy with no senatorial ancestors, but by Caesar's day it was really only used to describe those who obtained the consulship--particularly those who, like both Cicero and Marius, had had no senatorial ancestors of any kind, yet got themselves a consulship anyway). Mind you, within the nobiles, at least half (and probably more) of the nobiles gentes were plebeian--the Lex Licinia, which was based on an older tradition, in 367 guaranteed that at least one of the consuls for each year by law had to be a plebeian, and starting in 172 we see consular years in which both consuls are plebeians. In terms of social status, that's where the class lines lay in the Republic--those who had senatorial ancestry (and within that those who were nobiles) and those who did not, with the equites being a funky little subset of those who don't.

By the Principate, mind you, things change completely. The patrician order essentially ceased to exist by Augustus' time, having been wiped out during the civil wars. Augustus artificially recreated the patrician order by enrolling many plebeian nobiles and promoting the very wealthy, but the manipulation of the senatorial class (and the equites as well, whom Augustus officially added as a legal class--before they had been merely an unofficial subset of the plebeians) by the emperor meant that many of these social distinctions didn't mean much anyway, and they were politically neutered so it doesn't really matter all that much--you'll see huge numbers of equites who are freedmen (and even freedmen patricians--by Juvenal's time these were probably quite common, actually), for example.