This is purely a prediction on my part. I did not read the books nor comic if the true identity was revealed, though I believe many of the mysteries in the series is best to be left unexplained.
First, an analysis.
Village Chief is someone of great influence, able to influence the Minister of Development against the biggest developing firm, Ichinomiya. He has access to high quality and bizarre items (his fanned suit) and great wealth (enough to compensate for whatever the Ichinomiya group was chipping in for the minister), but he somehow chose to seclude himself from the society.
He does not have a particular tone besides random kappa related statements, seems not to have been raised in a particular way, and is well accustomed to the society when required to. He is well respected in the under bridge and regarded by everyone, especially sister, as a dependable person. The two times we've seen him in his usual self, however, he seemed particularly skillful in infiltration (coming in behind Shimazaki in a supposedly off limit office without her realising) and intimidation (threatened to kill). Threaten to kill is a peculiar trait as it is vastly different from Ichinomiya's way of intimidation via social and financial impact. It shows that he does things differently from politicians and businessmen. At final speech of Minister of Development, when asked by his subordinate who is he, he answered, "It's better for you not to know". Of both the time he appears, he wears a kimono. During season 2, we also gain insight that he is about the same age group (middle age) as Shiro (discussion about health and argument) and he is bad with group work (refuse to stay a week with the team, not a team player, more of leader or someone who simply gives orders rather than receiving them).
So there, Village Chief is/was a wealthy, influential person. He has great leadership, is mostly a lone wolf and does not shy away from intimidation by death. He also does not have speech traits, is well accustomed to society and wears mostly a kimono.
Remember Jacqueline when Billy did not come for their anniversary? She changed into a kimono when decided to head back and kill the head of the bird gang leader if something were to come to Billy. Kimono is something that Japanese mafia are obsessed with.
If I am to guess, he is/was the head of a big, dangerous mafia gang which has lots of financial background and ties on different background. Its influence and financial ground is at least on par with Ichinomiya group. I couldn't predict why he left, there is just too many possibilities without any ground to go on. He could be retired, betrayed, still on a relatively hands-off business with the gang, mafia dissolved, etc etc....
