Much of our modern cultural appreciation of Shakespeare has its roots in the 18th century theatre. Of course, Shakespeare was successful in his own time, and after his death (1616) he was eulogised by contemporary poets including Ben Jonson and William Basse. However, he wasn't immediately regarded as a 'canonical' or even 'literary' playwright, but more of a popular one, until the early 18th century.
In the Restoration period (i.e. after the return of Charles II to the English throne in 1660), the revival of English theatre included performances of Shakespeare's works, although these were less immediately popular than the collaborative works of Beaumont and Fletcher. Between the works of the latter, and of Shakespeare, they formed a useful, ready-made repertoire even though they had been written over 50 years earlier, as the theatre had withered away during the Civil War and the 'Interregnum', when Cromwell ruled as Lord Protector and Puritan censorship was heavy.
However, these performances took significant liberties with Shakespeare's texts, in a way which we wouldn't recognise today. Musical interludes and dances were inserted, scenes were cut, and lines altered to make topical political points. In 1681, there was even a version of King Lear staged with a happy ending, which rather undermines the entire point of King Lear.
In the early 18th century, literary connoisseurs and professional playwrights and directors (or 'theatre managers' as they were generally known) were keen to preserve and celebrate Shakespeare's works in their pure and original form, and revive them for a modern audience. An important figure in this movement was John Rich, the manager of the Theatre Royal in Covent Garden, who is on record as having lobbied for a permanent monument to Shakespeare from the 1720s. Eventually, with funds from notable cultural figures such as the 3rd Earl of Burlington and the poet Alexander Pope, a elaborate marble monument was erected, complete with life-size marble statue of the playwright, in Westminster Abbey in 1741. This was reproduced in smaller terracotta and porcelain versions, aimed at a wealthy and educated audience, and further established the playwright's image and significance in the public consciousness.
After Rich, and perhaps more important in terms of Shakespeare's reputation, came David Garrick: a theatre manager, a (good) actor and (bad) playwright. Garrick was a significant celebrity, moving among aristocratic and artistic circles; and his mission to champion the works of Shakespeare played a crucial role in bringing those works to an even wider popular audience, beyond the literary elite. Garrick's reputation as a Shakespearean actor was so great that, among the many portraits of him produced in his lifetime, one of the most important (and largest) was a canvas by the painter William Hogarth, showing Garrick in character as Richard III.
So, by the 1750s and 1760s, Shakespeare's theatrical reputation is more or less set. Interestingly, Shakespearean associations become something of a standard for successful actors having their portraits painted - for example, the tragic actress Sarah Siddons, whose most celebrated role was Lady Macbeth, was portrayed as the emblem of tragedy by the fashionable portraitist Sir William Beechey in 1793, complete with a marble monument to Shakespeare in the background.
One further, significant development in the construction of Shakespeare's reputation, particularly when it came to getting his works into the hands of schoolchildren, was the 1807 publication of The Family Shakespeare by Dr Thomas Bowdler; a physician who believed that children should be able to enjoy the plays without accessing any of the sexy or violent bits. In fact, most of the child-friendly adaptations were probably prepared by his sister, Henrietta Bowdler, but for reasons of 'respectability' it was published under his name. While Bowdler's efforts were mocked by some contemporary critics, it meant that for the first time, Shakespeare's works could reach a younger and a wider audience, meaning that the plays became not only a standard of English theatre throughout the 19th and 20th centuries, but also a standard educational text.
/u/OwlOfDerision did a great job talking about how Shakespeare became one of the greatest writers in England. However, his international reputation was by no means secure by 1750...you might almost say, what international reputation. From an earlier answer of mine on how Shakespeare became "the best playwright ever":
Shakespeare's enshrinement as the preeminent literary voice of England was secured in the first half of the 18th century. Up until the point, he had enjoyed basically zero international reception. Voltaire, in his obsession with English post-Restoration culture, is usually considered to have "introduced" Shakespeare to the wider European literary elite. But his 1733 assessment was ambivalent:
a fecund genius, full of vigor, ranging from simple naturalness to the sublime, without the least glimmer of taste or the slightest knowledge of the rules.
Indeed, Voltaire was toeing the 1600s English party line, the neoclassicizing Age of Reason. It was not until the nascence of romanticism in late 18th century Germany that critics and authors would claim Shakespeare as the paragon of a literary world that transcended national borders. To Gotthold Ephraim Lessing, the much maligned ghost of Hamlet is a true character that made audiences believe in the enchanted world again, even in an age of reason. Johann Wolfgang von Goethe insists that if Shakespeare's dramatic logic appears illogical, the fault is in us, not the play: we need to insert ourselves into his world. Works of art make their own rules; we need not apply external ones.
These German proto-romantics' use of an English author, instead of a native one, as their exemplar by the turn of 1800 completes Shakespeare's rise from upstart crow to rustic talent to English paragon to international superstar.