This is a complex story narrated by a barber who doesn't understand the significance of his own observations and report. He is a reliable narrator in that he has a perfect recall of events and words spoken. Yet he is an unreliable narrator in that he does not understand the events that were carried out right in front of him.
The theme might be expressed by the Biblical quote: "many that are first shall be last; and the last shall be first" (KJV Matthew 19:30).
In other words, the theme is that the least important person in town, Paul Dickson, turns out to be the most important, and the most important person in town, Jim Kendall, turns out to be the least important. This theme can also be described as appearance versus reality: Jim gave the appearance of superiority to the foolish people around him while Paul gave the appearance of foolishness, yet he was the only one to see reality and act accordingly (though illegally).
The conflict might be best expressed as Human against Human and is represented most keenly as Jim against Julie Gregg, yet he is also against the whole town as he manipulates, beleaguers and intimidates the whole lot of them, treating none of them better than he treated his wife, while seeing them all through his manipulative cruel sense of humor.
Well, [Jim] didn't have no intentions of bein' [at the circus] or buyin' tickets or nothin'. He got full of gin and laid round Wright's poolroom all day. His wife and the kids waited and waited and of course he didn't show up.
The ultimate expressions of the conflict come in Paul's conflict against Jim and in Doc Stair's conflict against both Paul and Jim: Paul sees the wrong that Jim has done and acts against it; Doc Stair sees the wrong and sees Paul's resolution redressing that wrong and decides how to report it.
[Paul] was shakin' so hard that he couldn't control the gun. He let fire and Jim sunk back in the boat, dead.
Doc Stair, bein' the coroner, ... examined the body and said ... [there] was no use ... callin' a jury, as it was a plain case of accidental shootin'.