Ultimately, everybody has an agenda.
Remember that "history" in the modern almost journalistic sense of the word was not really even a concept at this time. The point was not to convince "scholars" but regular people.
So, who then are the contenders as "historians?"
Tactitus gave us the first mention of the Christians, and provides external evidence of Pontius Pilate as the procurator of Palestine who had Jesus executed. His agenda in writing, however, was mostly to vilify Nero, blaming him for the destruction of Rome in 64 AD. Nero, of course, had alrady blamed the Christians - so Tactitus' agenda would be to exculpate the original scapegoats. By being non-Christian himself, however, he answers modern charges of being too interested a party.
Flavius Josephus also mentioned the Christians, but again, really only in the course of his agenda. Josephus was a nothing more than a traitor to his own Jewish cause. Although he fought with the Jews, when he was caught, he turned traitor. Eventually becoming a Roman citizen, he wrote a story about the war that modern historians find to be questionable in some places - but again, because of his agenda. In passing, however, he too mentions the Christians, but his agenda isn't really focused on them.
The Gospel writers were all eyewitnesses to Jesus. Did they have an agenda? Sure! The saw a dead man become not a dead man. They heard his message, and their lives were transformed by it. Of course they are biased! But, and this is a big but, they had the most most information about Jesus. Their agenda is to make Jesus known, and they all died really horrible deaths to do that. By modern standards of 'history,' their primary accounts would be discounted because they are too close to it.
Mark (who studied Peter) had probably written down his Gospel by 68 AD (before the Temple was destroyed), Luke (who studied under Paul) by the early 70s, and Matthew, who was an eyewitness and disciple by the mid 70s. John's Gospel may have been the last - written at the latest, by around 95 AD. Each of these was an eyewitness.
Later Christian "historians" like Papias, Iraneus, and Tertullian were all Christian bishops who based their histories on the Gospels' record. Papias is the earliest of the three, writing in the early 100s. Tertullian lived from about 160 - 225, and Iraneus (Bishop of Lyon) lived about the same time, dying in 202. Each of these men (and Eusbeius, who should also be in the category as well) lived well after the events of the New Testament. These men saw the Gospels as primary accounts, and wrote their own works as secondary ones.
Later historians, using many sources - not just the Gospels - compiled their accounts in later centuries. They never claimed to be eyewitnesses. But, they also never claimed the four canonical Gospels were fundamentally more "accurate" than any of the other sources. Canonization wasn't even really discussed until Iranaeus - and Iraneus was the one saying that these four canonical Gospels were the best sources he found.
Even today, a primary account of an event that transforms a person is going to be biased. Ask a New Yorker about 9/11, and there will be a bias. Ask a Palestinian, or a member of Al Qaeda, and it will be equally biased. Indeed, I would argue that every account will be told from a certain perspective. Was Mohammed Atta a terrorist? a common criminal? or a martyr for Muslim cause? There is no way to be neutral, nor is there any need to be.
The point, then, is not to ask "who to trust," since none of these accounts directly conflict. The only difference is in the fullness of the information (hint: the Gospel writers tell you a heck of a lot more about Jesus), and the agenda. To ask, "Are Tacitus and Josephus more 'unbiased'?" is thus still a flawed question, because in the end, every account has a bias. The question is, are the biases explicit, and do they interfere? Neither Tacitus nor Josephus really have much bearing at all on this.
And the Gospel writers? Of course they are biased. That's the point. They were transformed! They had a story to tell. The only question is, do you believe their stories? I know that my life has been transformed in the same way by the same man who died and then came back. Personally that is the kind of evidence I need.