Although there is already a question on how judges were appointed during the period after the First Temple was built, that question did not cover the period of the Judges, following the time of Joshua. How were they appointed? Did they have to be prophets, as Devorah was? Were they appointed by committee or by a single leader, or by concensus? Were there other women judges? And was the appointment of a woman, Devorah, controversial in its day? I don't know if there answers to any of the above, but I find the questions fascinating.
Rabbi Shlomoh Aviner writes this in regards to how Devorah was appointed:
Her unique appointment is explained by the Tosafot in the following way: 1. She was a prophetess who received a unique prophetic ruling (Tosafot on Niddah 50a). 2. She was willingly accepted by The Nation of Israel for this reason (Tosafot on Baba Kamma 15a). In fact, an individual who is usually unqualified to be a judge can be accepted as one for a special reason if both sides of a dispute agree. In a rare case, even a family member of one of the sides, or a shepherd, who most consider unfit can serve as a judge in monetary (but not halachic) matters (Sanhedrin 24 and Chiddushei Ha-Ran on Shavuot 30a).
Many shoftim (in sefer shoftim) were chosen by Hashem, or an angel, or by becoming a prophet. It is explicitly mentioned for some others that they were strong, and therefore it would seem that they came to judge Israel because they seized leadership. Other times it just says "He judged Israel for [insert years] years, and then he died."
Because of the nature of the time period for Israel, that there was no centralized leadership and everyone kind of did whatever they wanted, it would seem that shoftim were not elected or appointed, but rather chosen by Hashem, or just came forth to save Israel in a time of crisis, which is the role of the shofet in the cycle throughout the sefer.