who passed through universities with radiant cool eyes hallucinating Arkansas and Blake-light tragedy among the scholars of war,
They did go to school. But it sounds like they spent much of their time "hallucinating."
The reference to "Arkansas," a southern state with a large rural population, contrasts with the big-city vibe we've gotten so far. They were just as interested in small-town America as they are in the city.
"Blake" refers to William Blake, the late 18th/early 19th century English poet and painter who specialized in prophecies and visions. We can't even begin to describe how wild his writing is – you'll have to check it out for yourself. See, e.g., "The Tyger" and "The Lamb" in Shmoop Poetry.
Not surprisingly, Blake was one of Ginsberg's favorite all-time poets. In fact, Ginsberg had a vision in which Blake came and read his own poetry aloud (Source).
The best minds are hanging out in the university along with "the scholars of war," whom you could think of –from Ginsberg's perspective – as the "worst minds."
Ginsberg believed that universities were responsible for perpetuating warfare by inventing new weapons, supporting conservative ideas about the necessity of military power, and by generally contributing to the capitalist system that he hated. Despite these strong opinions about universities, Ginsberg didn't dislike all his teachers at Columbia. His favorite course was a seminar taught by the famous literary critic Lionel Trilling (Source).
who were expelled from the academies for crazy & publishing obscene odes on the windows of the skull,
They were kicked out of school ("the academies") for being "crazy" and for writing poems filled with obscenities.
They published these "odes" on "the windows of the skull," which we take to mean eyes.
This line also makes direct reference to Ginsberg's life. He was expelled from Columbia for writing "obscenities" on the window of his dorm room (Source). He later returned to finish college, but the experience left him with bad feelings toward academia.
who cowered in unshaven rooms in underwear, burning their money in wastebaskets and listening to the Terror through the wall,
In addition to all their other psychological problems, they were extremely paranoid. They were cowering from "the Terror."
Everything is mixed up, and it's the room, not the person in it, that was "unshaven." Just go with it.
They were burning their money in the trashcan as though they wouldn't need it anymore. Which you probably wouldn't if you knew that something called the "Terror" was on the other side of the wall.
who got busted in their pubic beards returning through Laredo with a belt of marijuana for New York,
A lot of "hipsters" grew long beards, which clashed with the respectable clean-shaven-and-crew-cut look of the 1950s. In the 1960s, as you probably know, long hair became definitely associated with the hippie sub-culture, which grew out of the Beat movement.
Back to "the best of minds": they tried to smuggle marijuana from Mexico, where it was cheap and plentiful, to America through the border at Laredo, Texas. They planned to bring it to New York to sell to friends, but they "got busted."
who ate fire in paint hotels or drank turpentine in Paradise Alley, death, or purgatoried their torsos night after night
It's hard to know just how many references to drugs this poem contains unless you have an encyclopedic knowledge of every slang term for drugs used in the 1950s. It's possible that "ate fire" is one such reference to drugs. The point seems to be that they put very unhealthy substances into their bodies, like "fire" and "turpentine."
The speaker may be exaggerating here, as turpentine in particular is a toxic substance that can be used to dissolve paint.
We wish we could tell you exactly what's going on, but you'll just have to go with the loose connections the poem offers.
Paradise Alley was a slum in New York, and you can bet that the hotels there were cheap and run-down.
In Dante's Divine Comedy, "Purgatory" was a mountain that stood between heaven and hell. People in Purgatory were technically saved, and therefore destined for Heaven, but they had a lot of problems to work out first.
The people in Dante's Purgatory keep repeating the same pattern of mistakes over and over again, be it envy or lust. Similarly, the people in Howl "purgatory" their chests by repeatedly putting bad things into them, like drugs and other stuff that we'll learn about in the next line.