This depends on what constitutes a "legit Pope" in your view. The topic is rather complex, so I'll try to address different issues regarding what constitutes a Pope.

To begin with, the scholarly consensus is that Jesus and St. Peter are real historical figures (their spiritual or religious status is outside the scope of this discussion). St. Peter was the bishop of the Christians in Rome, and so from the standpoint of the Pope being the Bishop of Rome, St. Peter is the first Pope. This is the teaching of the Catholic church as well.

However, the history is not quite as simple as all that. To begin with, the term "pope" wasn't used among the earliest Christians as far as we know. The earliest examples of the term have it being applied to priests (primarily in Eastern, Greek speaking areas) and bishops (in Western, Latin speaking areas) in the 2nd century (see: http://www.newadvent.org/cathen/12260a.htm). Tertullian specifically addresses this in On Modesty. This isn't surprising as the term "pope" is "papa" in Latin and "pappas" in Greek - in both cases, an informal word for father, almost as papa or dad would be used in English today. So from that perspective there were many "popes" early on and the real issue is the aspect of primacy of the Bishop of Rome.

That aspect of primacy is taught a more thorny subject. There are certainly early examples of other Christian communities looking to Rome for example or teaching (for instance, Clement I's Epistle to the Corinthians, found in some early Greek bibles, or Victor I's attempt to resolve the date of Easter during the 2nd century). Many other Roman Bishops made similar claims through the centuries but they were not generally acknowledged in the Eastern churches. However, this was also the case with the other apostolic Patriarchates, collectively referred to as the Pentarchy. These were Rome, Constantinople, Alexandria, Antioch, and Jerusalem. In particular, Jerusalem (whose first Bishop was St. James) was regarded as the seniormost of the Pentarchy, but this role was gradually eclipsed by Alexandria due to Roman persecutions of the Jews in the first and second centuries. The fifth century Council of Chalcedon issued an edict that the Bishops of Rome and Constantinople (New Rome) were equal in status. Later, at the end of the 6th century, the Patriarch of Constantinople was dubbed the "ecumenical patriarch", interpreted by some as making the Patriarch of Constantinople the primary authority in Christianity. In the seventh century, Jerusalem, Antioch, and Alexandria all fell during the Islamic conquests, which led to them falling out of prominence and left Rome and Constantinople as the most important remaining Christian centers. After this point, the Roman bishop can claim with more justification to be first among all the bishops, though this is never accepted in the Eastern, Greek speaking world and in part leads to the schism in the 11th century. Pope Leo IX was the pope at the time of the schism and made the case to the Patriarch of Constantinople that Rome was the "head and mother of all churches", a claim in part based on the strength of a document called the Donation of Constantine, which is now regarded by historians as a forgery (in fact Pope Pius II wrote on the subject and came to the conclusion that it was a forgery, albeit before becoming Pope).

The pope is also considered to be the pontifex, or pontiff, a title that existed before Christianity and originally referred to the priests of the Roman religion, with the head of the college being the pontifex maximus. This was a title held by Julian Caesar and subsequently the Roman emperors until Gratian relinquished it in the late 4th century. The Bishop of Rome started using the title in the 5th century, with the first example being Pope Leo I.

On the matter of the unbroken line, the Roman Catholic Church does trace such a line, but the status of who is actually in the line has changed over time. Various emperors (both Roman and Byzantine) have intervened to appoint bishops of Rome in the past, some of who are regarded as popes and others who aren't. There are also assorted other antipopes and so on along the way. For instance, the current Pope emeritus, Benedict XVI uses that papal name, but is actually only the 15th Benedict recognized as a Pope by the Catholic church. Benedict X (elected during the mid 11th century) is currently considered to be an antipope, but was not considered an antipope during the early 14th century when Benedict XI became pope. There were also three claimants to the papacy using the papal name Benedict XIV, but the only one that is recognized as a pope was pope during the 18th century.

2nd Timothy mentions greetings from someone at Rome named Linus; it's been speculated that might be the same Linus who's listed as the second pope.

Clement, who's listed as the fourth pope, wrote two letters that survive. Neither of them mention anything about the author or his office, though, except that he's a Christian named Clement living in Rome.

why is their such controversy regarding the existence of Jesus Christ?

it's probably useful to point out (though not sure if this is the place) that the whole "jesus didn't exist" line is much much much more popular in places like /r/atheism than in academia/scholarship. it's not a mainstream view.