I'm not sure what you mean by 'secular'; Josephus describes himself as an observant Jew from a priestly family, which is exactly how he comes across in the numerous works he authored (though his life experiences were very unusual for a Jew from Judaea of his time). He was a near-contemporary of Jesus rather than a contemporary, since he was born in AD 37, after Jesus' death. As for 'credibility,' that's awfully hard to summarize concisely, but yes, he'd generally be considered fairly reliable by the standards for historians of his day--which is to say, he's a useful source for events during and close to his lifespan, but the further back in time he goes, the heftier the spoonful of salt one must take his words with, as he relies increasingly on Jewish religious sources alone for "documentation."
At any rate, the specific five-sentence passage you're quoting--which is so famous in Christian history as to warrant its own name, the Testimonium Flavianum--has been highly controversial since the 17th century concerning whether it's original to Josephus, a later Christian scribe's interpolation, or a mixture of both (with the latter opinion currently prevailing among historians and theologians). I really don't know all that much about this debate. The first Christian writer to have clearly read Josephus' Antiquitates Judaicae (the work the Testimonium comes from) was the Church father Origen, who quotes from it extensively in his Contra Celsum, ca. 240 AD. However, Origen doesn't mention the Testimonium at all, though he does quote the very brief mention of Jesus found in Josephus' account of James the Just's execution, later in Antiquitates: "adelphon Iesou, tou legomenou Christou"--' [James] the brother of Jesus, who was called Christ'--which is indeed all that second passage, the only other reference to Jesus in Josephus' works, has to say about Jesus, based on all known manuscripts and citations of Antiquitates. Origen also disapprovingly writes of Josephus "kaitoi ge apiston to Iesou os Christo"--that he did not believe in 'Jesus as Christ.' Since the Testimonium's third sentence specifically proclaims 'Jesus was the Christ,' and also has much more (positive) to say about Jesus than that fleeting second mention, it's hard to account for why Origen doesn't cite it, especially since he quotes another passage (unrelated to Jesus) from the exact same chapter the Testimonium appears in.
The bishop Eusebius of Caesarea was the first Christian writer to quote the Testimonium in its present form, ca. 320 AD, in his Historia Ecclesiastica. The church father Jerome quotes it as well (392 AD, in De viris illustribus) with the difference that he quotes the third sentence as "Credebatur esse Christus" ('He was believed to be the Christ') rather than "He was the Christ." Three further works--an anti-Jewish apologetics text by the anonymous "Pseudo-Hegesippus," written probably in the late 4th or early 5th century; the Byzantine bishop Theoderet of Cyrrhus’ 5th-century AD commentary on Daniel; and the twelfth-century Syriac patriarch Michael the Syrian’s Chronicles--quote the Testimonium exactly as Jerome did, all of them noting that Josephus was an “unbeliever.” After that point in time, pretty much all Christian writers are clearly quoting from manuscripts in which the Testimonium appears in its present form.
While there are Jewish manuscripts (usually in Yiddish) of Josephus by the High Middle Ages, they’re uniformly of poor quality and heavily adulterated with other materials drawn from unclear sources; it seems that later Jewish scholars only came to know Josephus through Christian writers’ frequent use of the Testimonium in anti-Jewish apologetics. This isn’t particularly surprising--Josephus wrote in the immediate aftermath of the destruction of Jerusalem and the enslavement or forced emigration of much of Judaea’s population to far-flung corners of the classical world, hardly the opportune moment to establish himself as a scholar among his peers; plus, he wrote primarily for a Roman audience anyway, as a Roman citizen and Flavian patron, considering himself an apologist for the Jews (not just in the religious sense). Furthermore, Josephus had actually been captured by Vespasian’s troops during the Jewish revolt, and as ‘probation’ later worked as a translator for the Romans, encouraging Jewish forces to surrender--which likely did little for whatever reputation among his people his priestly background and obviously high level of Jewish learning might otherwise have secured him. Therefore, the debate over the Testimonium’s authenticity has always been, and largely remains, a debate among Christian scholars.
In the late 16th century, Protestant writers began to challenge the authenticity of numerous sources frequently drawn on in Catholic writings, among them the Testimonium Flavianum. Initially, these challenges were quite heavy-handedly political in nature, premised not on actual textual evidence but dubious claims to the effect that no Jew would’ve had anything but vicious and hostile things to say about Jesus and Christians, therefore the entire passage must be a fake—ironically mirroring the same anti-Semitic attitudes in which Catholic writers’ citations of the Testimonium were often steeped (by way of example: another passage from Josephus that was ever-popular with Catholic sources briefly mentions a Jewish woman named Maria, who allegedly ate her own child’s body to survive during the Roman siege of Jerusalem; this became a favorite illustration of Jewish perfidy and ‘blood libel’ savagery) . By the mid-17th century, however, Protestant scholars had begun to point out the conflicting evidence from Origen, Jerome et al. for the Testimonium’s authenticity. The controversy continues to this day, and as I mentioned earlier, the prevailing scholarly theory at this point is that the Testimonium is probably partially interpolation, but not a wholesale fabrication. Critical study of the passage usually focuses on the phrases “if indeed one ought to call him a man,” “He was the Christ,” and “On the third day he appeared to them restored to life, for the prophets of God had prophesied these and countless other marvelous things about him.” Some scholars believe that those phrases are probably wholly interpolated, while others--pointing to the slight but significant difference in the third sentence’s wording found in Jerome and others—suggest that similar minor alterations may have been made to the other two problematic phrases, transforming what were originally Josephus’ observations about what Christians believed to be true into a seeming profession of what Josephus himself believed to be true. The problem, of course, is that even assuming a generally positive opinion of Jesus and Christians on Josephus’ part (which is certainly possible), it’s extremely difficult to read those particular phrases as anything less than explicit professions of Christian faith, and nothing else in the roughly one thousand surviving pages’ worth of Josephus’ works we have—which, again, mention Jesus only one other time, fleetingly and rather unremarkably: “Jesus, who was called Christ”—suggest that Josephus was anything other than a mainstream, if highly educated, Jew in his beliefs. Unfortunately, it’s unlikely that there’s much in the way of ‘smoking gun’ new evidence left to be found out there.