The phrase out of one’s skull means out of one’s mind, crazy.
The earliest instance recorded in the Oxford English Dictionary (OED – 2nd edition, 1989) is from The Electric Kool-Aid Acid Test (New York, 1968), by the American author and journalist Tom Wolfe (born 1931).
But I have found earlier occurrences of the phrase. The earliest does not mean crazy, but not part of a particular exclusive group (a sense not recorded in the OED); it is from the column Young Moderns, by Vivian Brown, in The Bee (Danville, Virginia) of 4th June 1955:
BOSTON YOUTH GOES SMOOTH; GETS WITH IT
Greater Boston’s youngsters are concerned with: 1. Getting with it and 2. being s-m-o-o-t-h. That’s the word from reporter Dick Sinnott who tell [sic] us: To qualify as being with it, a boy or girl must be one of the crowd, wear the right thing at the right time, and know every word of every song.
The girls are for the real s-m-o-o-t-h look. They’ll lounge at hen parties in blue jeans and over-sized plaid shirts, but in class or on a date they’re dressed to the teeth.
[…]
There aren’t many new expressions for the real gone lexicon but here are some:
Let’s loom outta the gloom—get with it.
He or she is out of his skull—not with it.
Yes-yeh—said quickly denotes affirmative.
Unit—automobile.
Go ape—a state of unrest.
Student prince—good looking boy.
Snooty cutie—a girl that’s real nice—but knows it.
How’s that grab you—what do you think?
The second-earliest instance that I have found means out of one’s mind; it is from Princess Needs Escort, by Mike Connolly, published in The Desert Sun (Palm Springs, California) on 11th September 1958:
Hollywood—The Greek Consul of Los Angeles is out of his skull trying to line up a suitable “royalty-type” young movie star to take Her Gorgeous 18-Year-Old Royal Highness, Princess Sophia of Greece, to this year Imperial WAIF Ball (sponsored by Jane Russell’s Women’s Adoption International Fund for Orphans) at the Beverly Hilton Nov. 22.
Sophia’s mother, Her Majesty Queen Frederika Louise of Hanover and the Hellenes, is all set with a date—Col. Elias Deros, aide-de-camp to King Paul of Greece—but not the Princess. Oh, I tell you, Ike’s headache with the Off-Shore Isles are nothing to compare with what we go through here in Hollywood!
Not recorded in the OED, the phrase to get smashed out of one’s skull means to get hammered. For example, this is the beginning of an article titled Drinking Law Archaic, Says College Paper, published in the Independent-Journal (San Rafael, California) of 1st November 1963:
Revamping of “grandparents” liquor laws to permit persons 16 or 18 to “get smashed out of our skulls” was called for today in an editorial in the Tower Times, college of Marin student newspaper.
The phrase bored out of one’s skull means beside oneself with boredom, bored stiff; the earliest instance that I have found is from the column The Lively World, by Milton R. Bass, in the Bennington Banner (Bennington, Vermont) of 19th July 1967:
The only ladies who do talk golf all the time are the single ones — this includes divorced and widowed — who see this as an opportunity to mingle with the men. They never seem to realize that the men are bored out of their skulls by the prattling and that she has become a bogeyman.
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