Shakespeare was brillliant. To understand just how brilliant he was, it helps to compare his works to those of his contemporaries, like Christopher Marlowe's Doctor Faustus, or The Devil's Charter by Barnabe Barnes. The difference in quality is quite striking.
Not all of his works are to my taste--I prefer his tragedies to anything else he wrote. And like all playwrights, his work is really intended to be performed rather than read. That said--I really do think he deserves his reputation as the greatest poet in the English language.
Shakespeare was there, and he compared his work to Christopher Marlowe's: The verdict was, Marlowe was good enough to imitate. A lot. As they say, imitation is the sincerest form of flattery. If it had been Shakespeare who had died at 29? Except that Shakespeare was far too conformist to have gotten into such trouble. In addition to his real excellences, that conformism is one reason why Shakespeare is held up as the ideal the way he is. If Shakespeare didn't think the quality was so strikingly different, I've no idea why anyone else would think so.
The notion that Shakespeare is the greatest poet in English is sort of peculiar. Frankly, Venus and Adonis has it over Marlowe's Hero and Leander mostly by being more ostentatiously hetero, and The Rape of Lucrece is a pill. Edmund Spenser's poetry was near as I can tell technically more accomplished. Even the sonnets are much more of a mixed bag than is often conceded. While some are indeed classic poems, others are labored, and a few are petty. But, part of Shakespeare's greatness is what he wasn't, namely, the disreputable Marlowe.
Partly Shakespeare played a role in forming the new national language (as opposed to regional variants further from London, like Spenser's.) But really, is Shakespeare's contribution so much greater than Geoffrey Chaucer's, who can fairly be claimed to actually have created English as a language capable of poetry? Shakespeare's claim to be the English national poet started with the history plays laying out the history of
"England," aka the Tudors. And when the English conquered most of the world, Shakespeare went along as the emblem of their cultural superiorty.
Also, part of it is that Shakespeare was not just a great poet, but a dramatic poet. Verse drama was a form that only briefly flourished. Temporary as it was, its grandeur (when successful,) gave it the role played by the epic in other literatures. It was a national touchstone. Verse drama has an elevated tone not shared by recent drama, despite the occasional attempt to revive it.
In this respect it is notable that Shakespeare was a pioneer in prose drama. I think that is because part of Shakespeare's greatness as a poet was demotic. He heard the poetry in common speech. The really striking thing about Shakespeare was the relative lack of influence by Latin and Greek drama and poetry.
Despite the popularity of a handful of the tragedies, I still think that it is the comedies that really display Shakespeare's genius. The comedies really are rooted in human nature in a way the tragedies aren't, plus they are mostly free of the Tudor/monarchist rah-rah that peeps out of the tragedies.