I would add that when Shakespeare got started in the London theater scene, he was trying to make money and elevate his family. The acting and, more importantly, providing plays for the Lord Chamberlain's Men, was a business. And he did well: not only was he able to buy the best house in his native Stratford, but also a coat-of-arms for his family. In Shakespeare's day, "serious" poets didn't write plays; they wrote lyric poetry (sonnets or non-dramatic poems like "Venus and Adonis").
That said, I think we can point to 1595 as a turning point. Around this year, he wrote Richard II, Romeo and Juliet, and A Midsummer Night's Dream--all masterpieces. While before this he was already popular enough to incur the jealousy of other playwrights (i.e., Robert Greene), around 1595 he started cranking out the masterpieces. While not every Shakespeare play is a masterpiece, over a dozen (at least) are.
Although there are all sorts of issues with the various texts of the plays, thankfully we have versions to peform and read, so Shakespeare's fame has not been limited to his own time, but has only increased over the years.