Uncle Billy probably was plotting against the other outcasts from the moment they were banished.  The story doesn't specifically indicate this, but the reader knows that Uncle Billy should not be trusted.  

and "Uncle Billy," a suspected sluice-robber and confirmed drunkard.

He's a criminal and a drunk.  He doesn't exactly scream trustworthiness.  

The specific point in the story when Uncle Billy definitely begins his plotting is soon after Tom and Piney arrive with horses and provisions.  Uncle Billy laughs at just about anything young Tom says, because Uncle Billy knows a lot of it is bluff and bluster.  Mr. Oakhurst gives Billy a swift kick to make him stop, and Uncle Billy goes away from camp to sulk.  While among the trees, Uncle Billy vents and sobers up.  

As it was, he felt compelled to retire up the canyon until he could recover his gravity. There he confided the joke to the tall pine trees, with many slaps of his leg, contortions of his face, and the usual profanity.

When Uncle Billy finally returns to the group, he discovers that everybody is enjoying each other's company.  He feels left out.  It's then that he plots his revenge.  

Suddenly an idea mingled with the alcoholic fumes that disturbed his brain. It was apparently of a jocular nature, for he felt impelled to slap his leg again and cram his fist into his mouth.

Three paragraphs later, Uncle Billy and the horses are gone.