The halacha states that 3 judges have to witness the person who goes into the mikva for conversion. According to Biblical law, are 3 necessary (assuming mikvah is needed) and is the conversion complete with only one judge post facto (assuming the other requirements are fulfilled)?
The Talmud states in several places (Yevamos 46b, Kiddushin 62b), based on Biblical exegesis, that 3 people are required to witness something in order for the conversion to be valid:
גר צריך שלשה מ"ט (ויקרא כד, כב) משפט כתיב ביה כדין
Conversion needs 3. Why? It says "judgement" by conversion, like a court case (which needs a Court of 3 judges).
Tosefos in Kiddushin discusses what the three need to witness.
נראה דוקא בקבלת מצות הוא דבעינן ג' אבל בטבילה בחד סגיא אם כבר קיבל עליו המצות וראיה מההיא דיבמות (דף מה:) דקאמר התם ההוא דהוו קרי ליה בר ארמאי פירוש משום דמל ולא טבל אמר ריב"ל מי לא טבל לקריו כלומר וכי לא ראה קרי מימיו דצריך לטבול א"כ משמע מדנפיק בטבילת קריו משמע דלא בעינן שלשה ומיהו קשה מהא דאמר (שם דף מז:) דמושיבין אותה במים עד צוארה ושני תלמידי חכמים יושבים מבחוץ ומודיעין לה מקצת מצות קלות ומקצת חמורות ופריך והא בעינן שלשה ומשני תני שלשה אלמא משמע דלטבילה נמי בעינן שלשה ויש לומר דודאי למצוה בעינן שלשה אבל לא לעכב אי נמי י"ל דהא דבעי שלשה היינו טעמא משום קבלת מצות
Tosefos first brings support that the three are only necessary for the acceptance of mitzvos, but that one is enough for mikvah. Then Tosefos questions this, and suggests that while 3 are ideal for mikvah, one is enough after the fact (but seemingly would still be necessary for accepting mitzvos).
The Rambam concludes that someone who immerses before 2 people is not a convert - Hilchos Isurei Biah 13:7:
טבל בינו לבין עצמו, ונתגייר בינו לבין עצמו, ואפילו בפני שניים--אינו גר
If he immerses by himself... even if it is infront of two, he is not a convert.
There is another question currently being discussed that touches on similar points.
First off, any conversion is done under the assumption of Rabbinic approval (i.e.: that the authority of the conversion rests upon the approval of the rabbis). This is similar to how all marriages are done "Kidas Moshe ViYisroel" with Rabbinic approval, which allows Beis Din to retroactively nullify marriage in certain cases. Famously, the Rabbis refused to accept conversion during certain periods of history regardless of the process of the convert.
Halachah demands that conversion requires kabbalas ol mitzvos, a small amount of learning, tevilah in a mikvah in front of beis din (though there are ways to circumvent this one - namely if they turn around until the convert is under water), and (for a man) circumcision. No "biblical" conversion is acceptable without these factors, because that is what is required by the rabbanim.
Aside from all that, the Tanach is largely quiet on what constitutes "conversion." It is clear that circumcision is definitely required, but it isn't readily apparent that mikvah for a woman is a d'orayta requirement, and it certainly wasn't required to be done in front of Beis Din.
For the clearest example of this, look at Sefer Rus and the associated gemaras. At no point does Rus or Orpah engage in what we would consider an "official" conversion, even when married to two Jewish men. Rus declares her acceptance of the mitzvot, and Chazal consider her the "first ger/giyoret."
When they return to Eretz Yisroel, no one questions whether her conversion was valid. The only point in doubt was her previous status as a Moabite woman, which might prevent her from marrying into the Jewish lineage (though she would still be permitted to converts and mamzerim). The Chumash says "Lo yavo ammoni umo'avi bikahal Hashem." Boaz laid claim that the possuk be interpreted "Mo'avi vilo mo'avis," meaning that Rus, as a Moabite woman, would still be permitted to him. But there was many who questioned whether this was a savara subject to refutation or the de facto halacha. This was why Ploni Almoni was fearful of doing yibum on Rus. The halachic question wasn't even resolved until hundreds of years later when Shaul wanted to know whether he was bound by his promise to marry off his daughter to Dovid, the descendant of Rus the Moabite. After much back and forth, Amasa came forth and stated that it was a Halacha LiMoshe MiSinai, which isn't subject to logical refutation - Dovid was completely Jewish.
What we see is that Chazal accepted Rus' declaration of kabalas ol mitzvos as a complete conversion in their time, as this was the only "biblical" requirement. In the present day, there is no distinction to be made between the "rabbinic" conversion process and the biblical one, since the Torah explicitly states an obligation to listen to the Sages.