Our first description of Roger is short and accurate: "blunt, arrogant" (1.45). Roger is only concerned with moving up in society, and he doesn't care how transparent his moves are. He's a walking definition of fake it 'til you make it, with an emphasis on the fake. Roger is the guy with no actual friends, only sources, leads, and business contacts. As his Major dad says, "In my day I don't think we ever felt the need to 'work' our social contacts in such a manner" (13.152).
Roger's role in the book is to rack up as many offenses to British country customs as possible. Here are a few highlights:
The Major just can't even with all this buffoonery. With his wife and brother dead, his son is the only family he has left, and his son is terrible. "He and Roger seemed to have little common ground. […] In the shrunken world, without Nancy, without Bertie, it seemed very sad to be indifferent to one's own son" (13.214).
The Major does what he can to stop Roger from acting like the brat he has become. When Roger asks to borrow the Churchills, the Major flat out tells him no. But when Roger ditches him on Christmas, the Major declares, "He is no longer my son" (19.80).
Eventually the two make up, although it's less about acceptance and more about the Major just not caring anymore. He gives Roger one of the guns because he realizes the guns don't matter, not because he suddenly cares about his son. In the end, the Major "wondered again whether he had failed Roger" (25.81), but he just has to deal with it.
When we last see Roger, he's being his Roger-y self, fiddling with the musicians' instruments at the wedding as if he knows better. Maybe he's trying to get a gig out of it.