I think it's clear that Shakespeare didn't set out to become what he became--a famous writer. It was a perfect storm, of sorts, which thrust him into prominence. It was a period of great resurgence for the arts (the Renaissance), the Queen of England was committed to patronage of the arts (and Shakespeare's troupe, in particular), and Shakespeare was a writer with something to say about things people were interested in hearing. He was in the perfect environment for everything he did; despite that, he's significantly more popular today than he ever was in his own time.
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.
Shakespeare's intentions were first to become an actor. He left Stratford-on-Avon to begin a career on stage. He worked his way up the ladder in the acting profession: first as a gopher, then a prompter, and finally an actor. Luckily, in 1593, the Bubonic plague closed the Theatres of London, and so Shakespeare turned to writing long poems and drama full-time.
According to Enotes:
The next mention of Shakespeare is in 1592, when he was an actor and playwright in London. His actions during the seven-year interim have been a matter of much curious speculation, including unproved stories of deer poaching, soldiering, and teaching. It may have taken him those seven years simply to break into and advance in the London theater. His early connections with the theater are unknown, although he was an actor before he became a playwright. He might have joined one of the touring companies that occasionally performed in Stratford-upon-Avon, or he might have gone directly to London to make his fortune, in either the theater or some other trade. Shakespeare was a venturesome and able young man who had good reasons to travel—his confining family circumstances, tinged with just enough disgrace to qualify him to join the disreputable players. The theater was his escape to freedom; he therefore had strong motivation to succeed.
As far as we can tell, Shakespeare's early works were most likely collaborations with other playwrights and possibly actors. Interestingly enough, Richard III and Henry IV seem to have been some of his earliest works. Maybe he started with histories because there was less for him to make up.