Typo. His surname was Kinskey. My bad! 
The world had too little of Robert Shaw before he left it; whether as a king, an assassin on a train, an Irish mobster, or a vengeful shark hunter.
Charles Napier was great in The Blues Brothers, The Cable Guy, and as Sterling Archer’s inept cancer doctor (among other roles).
Kathleen Freeman was the first person I thought of. She did SOOOOO much work in a ton of sitcoms in the 60’s & 70’s. I’m sure most remember her as the nun (Sister Mary Stigmata (a.k.a. The Penguin) on the first Blues Brothers movie.
Ned Glass – he was ubliquitous in the 1950s through the 1970s. He was Leopold W. Gideon in Charade, “Doc” (who owned the store) in West Side Story, and the Ticket Agent in Grand Central Station who tries to turn in Cary Grant in North by Northwest.
I second Michael Ironside. He’s been on my radar ever since he was the chief Bad Guy in Cronenberg’s Scanners. Every time he appears in a Veerhoeven film, though, he loses limbs.
Eugene Pallette – Rotund, gravel-voiced actor in the 1930s and 1940s. He played Friar Tuck to Errol Flynn’s Robin Hood in 1938 and was The Duke in Huckleberry Finn in 1931. He showed up in lots of small parts in that era. Impossible to miss. He wasn’t really a nice guy in real life, apparently. Very right wing and an early Survivalist, he retreated to his bomb shelter in Oregon.
Three who became famous as character actors.
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Jack Elam - Early in his career in action films was regularly the first bad guy killed just to make it clear who the good guy was (The good guy was the man who killed Jack Elam). He was in among many many others; Once Upon a Time in the Old West (He was the guy who catches a fly in his gun barrel), Support Your Local Sheriff (he was the ex-drunk deputy).
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Strother Martin - He became very identified with his roll in Cool Hand Luke where he said, “What we have here is a failure to communicate.” He also played in Butch Cassidy and the Sundance Kid. The Wild Bunch and True Grit.
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** Mildred Natwick ** She played mothers and aunts a great deal. She was Jane Fonda’s mother in Barefoot in the Park, Rock Hudson’s mother in Macmillan and and Wife and the woman who set her hat for Victor Mclaughan in The Quiet Man.
What was interesting was his career arc, where he started out playing a deadly killer, but then he grew a beard and was always comic relief.
S.Z. “Cuddles” Sakall. Chubby Hungarian actor best known as Carl the headwaiter in Casablanca. Usually played a kindly but gloomy uncle type.
ZaSu Pitts. One of the greatest of all commediennes. Held her own against W. C. Fields in Mrs. Wiggs of the Cabbage Patch. Usually played a dippy character; Mae Questel imitated her voice when she first played Olive Oyl. Legend has it that she had more of her footage cut from films than anyone else because she constantly got bigger laughs than the star. She was a tremendous dramatic actress in the silent days, too, with her performance in Greed.
There is a possible addition to this OP - A father/son team of the same character. I’m sure there are other father/son teams doing the same characters (possibly Lon Chaney sr./ jr., but senior was just too big to be considered just a character actor. The same could possibly be said about the Douglas Fairbankses.
Anyway, Alan Hale sr./jr.). Son could have stepped in for father in his character roles and seemed to on occasion until son became a featured player in Gilligan’s Island. Father often played a sidekick to Errol Flynn as opposed to son’s Bob Denver.
This is not meant as a hijack - just an homage to both the question and the profession of character actors.
The first person I think of in this regard is Eli Wallach (Tuco Benedicto Pacífico Juan María Ramírez in The Good, the Bad, and the Ugly).
Derek Jacobi. Played Claudius against Branagh’s Hamlet. Appeared on “Frasier” as a Shakespearean actor who thinks his talents are wasted on a science fiction TV show. I thought it was funny because he does Shakespeare IRL. Modern audiences would most likely remember him as being the King in the recent “Cinderella” live action movie.
Or Claudius in*** I, Claudius***. 
He was also the printer of forged documents in The OdeSSa File.
I wouldn’t pigeonhole Jacobi as a "character actor, " myself. He’s been in tons of parts. He was Lebel’s assistant in Day of the Jackal, the printer in The Odessa File, the modern Narrator in Branaugh’s Henry V. He narrates the Penguin Aufiobook of Robert Fagles’ translation of The Iliad.
Plus you left out his biggest role – he played Claudius in I, Claudius*
*(and not just that other Claudius against Branaugh’s Hamlet. In fact, Jacobi himself has played Hamlet)
I’ve always had a weakness for Frank Morgan. He started in silents supporting John Barrymore but would be best remembered as the Wizard of Oz.
Forgot that one!
Ben Johnson
Warren Oates
Sgt. Tyree to Sgt. Hulka - these two have run the gamut.
John McGiver. Senator Jordan in The Manchurian Candidate, Dr. Chumley in the 1972 remake of Harvey, Mr. Babcock in Mame, the Tiffany salesman in Breakfast at Tiffany’s, and many, many more. Any time you want a dignified older man with a hint of mischief, he’s your guy.
Harry Dean Stanton - Repo Man and Kelly’s Heroes are two that have benefited from his skill.
Excellent choice! He’s brilliant as Tuco. Only trouble is, if you recognize Eli Wallach from The Good, the Bad, and the Ugly you won’t recognize him in much else. I saw a few minutes of How to Steal a Million (released the same year) in which he plays a rather conservative, boring insurance man. I almost couldn’t believe it was the same actor.
Type up this reply once and would have sworn I posted it already. Perhaps the hamsters ate well today.
Tracey Walter – Miller in Repo Man and Bob the Goon from 1989’s Batman
Harry Dean Stanton – This guy’s been the grizzled old guy for a long time. His IMDB credits start at 1954
Dick Miller – The character actor’s character actor
Man, there are tons more.
Richard Jenkins Got an best actor Oscar nomination for the wonderful little film “The Visitor”. Luckily that opened more doors and he has been very busy since. I know him from The Cabin in the Woods, but he was also in Jack Reacher and Eat, Pray, Love.
John C. Reilly A supporting actor Oscar nomination for Chicago. His rendition of Mr. Cellophane was the best thing in that movie IMHO. Probably people will know him from The Perfect Storm, Gangs of New York, Talladega Nights, and most any other Will Ferrell movie.