What is the relation between the characters in Fargo Season 2 and the movie characters?
I may have to be corrected for this but I do not believe that there is a direct plot or character link between Fargo (1996) and Fargo (TV Series) Season 2 (2015), they are only related by association through Fargo (TV Series) Season 1 (2014).
With regards to plot, the first series is related to the film through Stavros Milos's discovery of the money buried by Carl Showalter in the film. The first season of the TV series is then of course connected to the second though the Lou Solverson character, who refers to the events regarding 'Sioux Falls Massacre' which we subsequently see in season 2.
Thematically, all of the pieces are connected by the themes of bungled cover ups of murder, the intrusion of violent killers into what are otherwise close knitted communities and salt of the earth, dogged police officers who persist in doing good work despite the entrenched views of their (often incompetent) law enforcement associates - and of course, all of these 'true stories' exist in the same fictionalised universe.
While not a direct link to the movie, there is a big wink to the movie in at least one plot strain:
The first season has been the most directly connected to the events and arc of the film. Fargo's second round, however, did swipe one big chunk of the movie's plot, by masking a kidnapping victim in a burlap sack and tying him to a chair in a remote cabin with a malfunctioning old television set. The show even uses similar language to describe the mindset of the kidnappers, as an innocent bystander who encounters them says they're restless and bored … or, "Going crazy out there at the lake."
The article points out other elements:
There are many elements that tie the movie and TV Fargos together, but none moreso than the kind of character who always seems to set the plots in motion. On the big screen, it's William H. Macy's Jerry Lundegaard, a desperately broke car salesman who thinks he can get the money he needs by hiring someone to kidnap his wife. [...]
In Season Two, Ed Blumqist makes some terrible choices to support his ambitious wife Peggy (Kirsten Dunst)
Aside from lifting motifs from Carter Burwell's film scores, the show's first season more or less steered clear of copying the Coen brothers' actual music cues. But in the second year, the show loaded up on leftovers, serving up covers of [...] Jose Feliciano's "Let's Find Each Other Tonight" from the movie version of Fargo.
The article points out that the show also references other Coen brothers movies, e.g No Country for Old Men, The Man Who Wasn't There, The Big Lebowski,...