At the climax of a great Shakespearean drama, the once-great king surveys the ruins of his greatness. Betrayed by family and friends, most of his captains dead or treacherous, he will go mad. Indeed, he had always been borderline psychotic. Still he rants on, puffed with self-pity, promising revenge, alone in the gathering storm and the twilight of power. “I will do such things . . .” This is not King Lear, but Tony Soprano in the final series of a programme that changed the nature of television by bringing it closer to great literature.

The Sopranos is not Shakespeare. (Even Shakespeare’s claim to be Shakespeare is disputed by some.) But as television drama, it came closer to the sweep and heft of Shakespeare than anything before it, with its tragicomic themes of loyalty and betrayal, religion and violence, manhood and ageing. Here was an everyday story of Mob folk, with father fighting son, brother slaying brother, ambition at war with conscience, while they bought expensive, ugly televisions and watched their lives shrivel up from suburban angst. The Sopranos was about the competing demands of power and love: uneasy lies the head that wears the Soprano crown. Shakespeare would have loved it.

If Tony was King Lear, forever demanding of his courtiers and children “which of you shall say you love me most”, then Dr Melfi, his analyst in the series, was the king’s beloved Fool, telling him the truths he could not bear to know, and Meadow was his Cordelia, buttering up her Mafia dad while studying “morality, self and society” at Columbia University.

But Tony was also Macbeth, the ultimate warrior, a natural leader stooping to murder most foul (with a variety of blunt implements), and Carmela was his Lady Macbeth, ever more complicit in the crimes, but fastidious, demanding how to get the damned spot out of her Hermès scarf.

Here too was an Italian-American Hamlet, haunted by ghosts of the dead, betrayed by his mother, usurped by his uncle, wondering whether to kill them both before they whack him. In the background to the drama, the rival Mafia clans, Montagues and Capulets, fight it out over control of the disintegrating crime industry, disguising brutality in a cardboard code of honour.

Tony Soprano was the grinning, charming sociopath, who could smile and smile and be a villain: a Shakespearean hero, impossible to like but also hard not to love, conflicted and elemental and awe-inspiring: Coriolanus with a knuckleduster and a paunch. And what were Paulie, Sil and Big Pussy Bonpensiero if not Tony’s Bardolph, Peto and Falstaff? “Let me have men around me that are fat,” and boy, were those Sopranos fat, what with the capicola and ziti and Artie Bucco’s cooking. Like Shakespeare, David Chase, the creator of the series, revelled in food and sex, bawdy jokes, cruel satire and vivid metaphor. The perfect hit, says Uncle Junior, should be “as silent as a mouse pissing on cotton”.

The Sopranos demanded and deserved to be treated as high art. But if the programme’s ambitions were Shakespearean, its detail was Dickensian, an immense, sprawling 19th-century TV novel of ideas presented in instalments, with unforgettable cameos from people with names like Paulie Walnuts and Johnny Sack, and an entire coastline of cliffhangers. Victorian readers clamouring to discover the fate of Little Nell were no more avid than the millions tuning in to the last episode of The Sopranos, which was broadcast in the US this week.

In its relationship to viewers The Sopranos rediscovered an older form of cultural engagement. Like Dickens and Shakespeare, the series was not filtered through the opinions of lofty critics or a cultural elite, but swallowed raw, and then digested on a thousand discussion boards by millions of viewers: the audience hurled its bouquets or brickbats from the online pit. This was drama for the internet age, direct, immediate and brutal.

The Sopranos marked the moment when pop-culture television entertainment became high culture. It showed that the narrative strength of television can be harnessed to the central dilemmas of existence, and it did so not with ponderous self-consciousness, but with humour. Here was a genre ever ready to undermine and mock itself.

This was a tragicomic story of human frailty. A bad man with good in him trying to understand what makes him tick or, more often, explode. There has never been such a vicious parody of the American Dream, for at the end The Sopranos are where they were at the beginning: unhappy, dysfunctional, blind to themselves, slobbed out on the sofa in an unfeasibly hideous house, watching rubbish on a plasma screen and eating junk.

“You’re living in a f***ing dream . . . watching these jack-off fantasies on TV,” snarls AJ, Tony’s repulsive, insightful son, in the final episode. But as The Sopranos showed, fantasy TV can get real. Every scene came laced with ambiguity, between the characters, within the characters, and between Tony and an audience that found humanity in his sins. This is what distinguishes literature, which poses difficult questions without necessarily providing answers, from soap opera.

I am not going to give away the ending. Knowing how Hamlet ends does not detract from the play one jot, but nothing can compare with seeing a grand Shakespearean final scene for the first time. Suffice to say that after eight years of therapy and 86 episodes, as the end approaches Tony is still searching for the self-knowledge that has always eluded him.

As Paulie might have put it: “’Tis the infirmity of his age: yet he hath ever but slenderly known himself.”

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