There's a fascinating account of him by Paul Berman.
Cockburn is reminiscent of Christopher Hitchens - the English journalist who lives in America and writes stylishly about American and international affairs. The political framework may be leftwards, the cultural references English literature, quoted with ease to point the moral and adorn the tale.
Berman does explain why Cockburn was so indifferent about Israel Shamir's vile antisemitism:-
How systematically the man had gone after Israel, and how reluctant he was to say a word of criticism about the Soviet Union or the Islamic Republic of Iran or the terrorism of the anti-Zionists. Newfield took note of Cockburn’s hostility to Natan Sharansky, who in those days was a persecuted Jewish dissident in the Soviet Union.
... In his column he took to sniping in my direction, not always wittily, which other people in the Voice newsroom attributed to his distaste for the Jewish concerns that sometimes cropped up in my writings. He accused me of “pandering” to the Jews by writing about Holocaust denial and Noam Chomsky (a rich theme)
Cockburn shared the Israel (or "Zionism") obsession of the anti-imperialist Left:-
he does attribute 9/11 to “recent Israeli rampages in the Occupied Territories”
Alexander Cockburn woshipped his father, Claud Cockburn, who had lived an adventurous life where history was happening and who in the Spanish Civil War had the mouths of high-ranking Soviet officials in his ear.
Claud Cockburn was, in spite of appearances, a fine man who would never have turned over names to the Soviet police in Spain. But then, proud of his father, Alexander includes within the Wreck a brief memoir by Claud of his Spanish experiences, which leaves the impression that, in regard to the Soviet police activities, Claud was not a reluctant participant. About his friend Guy Burgess and the other Cambridge spies, Claud remarks that idealistic motives were at work, and these were “sensitive and informed” young men. Claud cites his own “experience in the field of espionage, or rather, counter-espionage.” He was “a section leader of the counter-espionage department of the Spanish Republican government dealing with Anglo-Saxon personalities,” which does sound like a job dedicated to informing the police. "
Berman believes that Claud Cockburn was a dark inspiration for George Orwell:-
I have always supposed that, when Orwell laid out the principles of totalitarianism in Nineteen Eighty-Four, one of his inspirations was Claud Cockburn, British correspondent: a cheerful example of a man willing to say everything and its opposite in the interest of a totalitarian state, committed to the renunciation of truth, to the hatred of free-thinkers, to the cause of persecution, and to the cult of obedience.
In Orwell’s portrait, the totalitarian mentality was never a matter of ideology gone awry, nor a matter of lower-class resentments. The mentality was a contempt for everyday morality and human considerations. It was a flippant nihilism, attached to no cause or principle at all, apart from love of tyranny. And, to be sure, no sooner did the Spanish anti-fascists go down to defeat than Claud Cockburn turned on a dime and set about composing justifications for Stalin’s pact with Hitler.
You could think this was a total damning of Alexander Cockburn. But Berman pays due attention to his elegant style and his love of anecdote, which at the end made me think I'd enjoy at least flicking through words of the co-editor of Counterpunch - though his constant jeering (in which he sounds like Hunter S Thompson), would get tedious after a while.