Thompkins was an important part of 'The Young Riders.' A 'regular' character, Thompkins served the traditional theatrical role of 'Everyman,' also known as the 'Chorus.'
This means that the character of Thompkins was used as an indicator character; he showed us the general temperament of the townspeople regarding issues. For example, his perpetual hatred of Buck was indicative of the way 'ordinary' townspeople would be expected to react. For many, the idea that 'The only good Indian is a dead Indian' was a truism which ruled their lives. Remember the 'Little House on the Prairie' books by Laura Ingalls Wilder? Ma voiced that idea on several occasions, though Pa seemed to be more tolerant - he listened to the advice of an Indian in 'The Long Winter.'
Another example of 'Everyman' is in the movie 'Evita' - Ché is used as 'Everyman' - check out how Antonio Banderas shows up everywhere, commenting on the proceedings as he goes. This allows the viewers continuity and a chance to identify with the character, rather than asking 'Who's that?' every time a new person pops up - and allows the script writers to get on with the story instead of having to supply each new character's background, temperament, etc.
William (Bill) Thompkins was the owner of Sweetwater's general store. Having a monopoly, he adjusted prices to take advantage of the people whenever he could (Home of the Brave). He was a civic leader (Daddy's Girl)with a prejudiced, bitter, and lonely soul. Nevertheless, he did have a kindly side to him, once you got to know him - witness the Christmas gifts in 'Star Light, Star Bright.'
He became bitter over the loss of his wife and daughter to an Indian attack. That hatred became so much a part of him that when even he found them alive he could not overcome it until too late, and so lost them forever (Pride and Prejudice). As usual, Buck was on the receiving end of his wrath.
In season 3, Thompkins followed the riders to Rock Creek to expand his store, much to their dismay. Still, he was a very valuable character to the show.
PS - If anyone's got a photo of Mr Thompkins, I'd love to link to it! Thanks, Kathy.