Just before the Christian era, many Jews are claimed to have migrated to Rome and other parts of the Empire as a result of economic hardship and continual wars between local powers such as Ptolemaic Egypt and the Selucids. It is even claimed that Jews were a part of Rome as early as 300-200BC. Despite this Jews do not seem to have been treated very well in the early/late Republic, evidenced by the fact that Hispanus, as Praetor of Rome in 139BC, expelled all non-citizen Jews from the city.

Many Jews visited and stayed in Rome for extended periods of time. It was very common for client rulers to send their children to Rome to be educated in the Roman style and as a form of hostage. A specific example relating to Jews would be Herod the Great, who sent his sons Antipas, Archelaus and Philip to Rome.

Julius Caesar and his immediate successors allowed Jews to practice their own beliefs without persecution. A popular name for this status is Religo licita (permitted religion), mentioned in the Apologeticum of Tertullian to describe the status of Judaism in the Empire. However this phrase is never mentioned in Roman Law, so it is unlikely that Jews were given formal rights, rather they were not persecuted because there was no reason for them to be.

We can see the attitude of the Romans toward Jews in the aftermath of the Jewish Roman Wars (66-135/later) in which the province of Judea was devastated, including much of the Temple of Jerusalem. After the destruction of much of Judea in 70AD, a 'Jewish Tax' or Fiscus Judaicus was introduced, requiring all Jews to pay around 2 denarii which had originally been paid for upkeep of the Temple of Jerusalem to now be paid toward the upkeep of Temples in Rome. A procurator in Rome was even appointed to specifically oversee the payment of this tax by all Jews in the Empire.

Now, on to your specific question about how you would rise in Rome as a Jew. Many 'successful' Jews were slaves in Rome, having been captured during the Judean Wars. This fact in itself immediately limits your ambitions as you will likely never be a citizen, only at best a freedman with no chance to hold any official position. A famous example of a Jew who many people know is Josephus, a scholar and historian among other things. He fought against Vespasian in the first Jewish-Roman war and was captured and enslaved by the Emperor. In 69 AD he was given his freedom by Vespasian, and eventually even granted citizenship and served as an advisor to Titus. Evidently it would not have been impossible for Jews to do well in Roman high-circles, but it would have likely been rare. His children were Roman citizens and although it is unknown whether they continued to practice Judaism, it would not be impossible that they continued to prosper as a relatively well-off trading family.

Paul the Apostle was claimed to be both a Jew and a Roman citizen, but some historians such as Helmet Koester have contested the evidence presented in The Book of Acts. Ultimately it is not impossible that he was both a citizen and a Jew, but there is little actual evidence for this. It is likely that most Jews in Rome and elsewhere continued to live as non-citizens, and there were very few Romans if any who would describe themselves as Jewish. This highlights just how difficult it would be as a Jew to prosper as anything more than a trader in Rome, as there are no examples of any Jewish-Roman aristocrats or Senators to my knowledge. Evidently virtually all Jews in Rome were non-citizens and part of the lower class.

Most Jews would have lived in Rome and traded just as any other citizen. They would have still been able to buy and sell goods, and it is likely that many were able to become wealthy and influential within their circles. However as mentioned before Jews would be forever excluded from citizenship in most cases, with Emperor Domitian even forcing non-practising Jews to pay the previously mentioned Jewish Tax. This explains the Roman attitude toward Jews quite well even if it was an isolated case; Jews were not openly persecuted (at least for very long, the tax seems to disappear around the 2nd century AD) but they were not considered Roman and would not be permitted to hold any positions of power. The perception of other Roman citizens toward Jews would ahve likely been very similar to the aristocrats, seeing them as practising a foreign and strange religion and being responsible for years of bloodshed in Judea. It is impossible to know how they were treated, but it is likely that they were treated as they have been virtually everywhere else for 2000 years - feared, scapegoated and unfairly taxed. If a Roman citizen was bringing charges against a Jew (who could not legally defend or prosecute anybody, since he/she is not a citizen) then it is likely that the Jew would not have been trusted and thus punished.

I hope my response answers your questions. In summary, living as a Jew in Rome would not have been very pleasant, just like everywhere else they have lived in ancient/medieval/recent history. A Roman citizen had many advantages over a Jew, and although there are some cases of Jews doing well for themselves, they are not common. Jews could prosper in Rome but they could never reach high office or hope to be treated like the other citizens. The best time to be a Jew would have been during the reigns of Julius and Augustus Caesar, as both relaxed religious laws and this seems to have continued into the future. There are no records of Jews marrying Roman citizens; Josephus married many times, but only to Jews or Greeks. Conversions would have been extremely unlikely, as it would be political suicide to give up your Roman religion in favour of a foreign and suspicious one.

-Adrian Goldworthy's many books on Rome, most importantly 'Augustus'

-Philip Francis Esler, Community and Gospel in Luke–Acts: The Social and Political Motivations of Lucan Theology

-Benjamin H. Isaac, The Invention of Racism in Classical Antiquity (ficus judaicus)

-The works of Josephus, including the Jewish War, which describe the tax imposed on Jews following the conclusion of the conflict

-Valerius Maximus for the early treatment of Jews evidenced by the actions of Hispanus

-Helmet Koester (article on his view of Paul)

-The Jewish Encyclopedia (I have used this in my descriptions of the early existence of Jews in Rome)