Ed Wood as the titular Glen or Glenda.
“Pull the string! Pull the String!”
Klaus Kinski in Aguirre the Wrath of God.
How about Jeremy Irons in “Dead Ringers”? That movie creeped me out more than anything else I’ve ever seen.
A classic, for me, is Vivien Leigh in A Streetcar Named Desire. I don’t know what’s so riveting-just the great acting, th fragility of it, or knowing that in real life she wasn’t too tightly bolted, either.
The first that came to my mind was Jack Nicholson in The Pledge.
Jack Nicholason and Louise Fletcher’s pied a deux in Cuckoo’s Nest.
Robert De Niro as Travis Bickle in Taxi Driver.
Julia Haggerty as Rita in The Wife
Harry and Kate (characters) in Angel Baby
Winner.
Helena Bonham Carter as Ophelia in Mel Gibson’s Hamlet.
I have to second Vivien Leigh’s performance, and add Kate Winslet’s Ophelia in Kenneth Branagh’s version of Hamlet. That woman is gifted.
Foot for Two? 
The headmistress in Picnic at Hanging Rock.
Are any of these examples of characters who are completely normal, happy, and functioning at the beginning of the movie, and totally off the deep end by the conclusion?
Fight Club’s Edward Norton isn’t happy, but he absolutely defines normal at the beginning of the movie. He’s just an average, unsatisfied office worker.
Oh, Harrison Ford in The Mosquito Coast
Not a whoosh, you are entirely correct. I was mixing my anti-war movies.
Jim Carrey in Me, Myself & Irene.
Beat me to it.
Speaking of Jack Nicholson - there’s The Shining.
And speaking of Kubrick there’s HAL in 2001: A Space Odyssey
Laurence Harvey’s meltdown in The Manchurian Candidate
Wilford Brimley in Alien qualifies. He was OK until he figured out that the entire human race was about an eyelash away from extinction because of what they’d dug up.
Whatchoo talkin bout Willis? Wilford Brimley wasn’t in “Alien”. You already posted about him in “The Thing” once.