The opening scene has an outlaw shooting a person for no real reason in a remote area. He leaves him to die. But he is still alive when Elmo Sippy comes by, but Elmo can't be bothered to help the man, and he dies. This is to establish that Elmo really has no conscience, but he hasn't had the opportunity to become a criminal. That is about to change.
In Dodge City, Elmo gambles and loses to the outlaw who shot the man he came across. The outlaw says he will kill Elmo if he doesn't come up with the money that he owes him. Elmo finds it easy to steal money from the general store's cash register when the owner is not looking. It's easy because the guy has no moral core.
This causes Elmo to go on a crime spree, robbing and shooting people in isolated situations, leaving no witnesses. Meanwhile, Matt has lots of anecdotal evidence that Elmo is responsible for the crime spree, but no evidence. He seems to find the entire thing hard to believe since Elmo has such a mild outward demeanor. But "nice" does not mean "kind".
Later, Elmo comes back to Dodge City to gamble once again with the outlaw whose demands caused him to go down a dark path in the first place. The end is ironic to say the least.
Matt suspects Elmo, trails Elmo, but ultimately has no proof because there was no such thing as forensics at that time. Today, if you zero in on the correct suspect, forensics can usually provide the proof.