Probably around 1587-8. It is known he left Stratford in 1585, after the christening of his twins, Hamnet and Judith, and surfaced seven years later, at Bankside, where he is either an actor, or a playwright. He is pilloried in a vicious attack on his character, published in 1592, by Robert Greene, a university graduate, who is irate and indignant at Shakespeare's 'uneducated' life, (Shakespeare did NOT go to University!). This is the first written reference to Shakespeare after his leaving of Stratford, and puts him firmly in London. However, he has already written a number of plays for the Queen's Men, the most senior troupe of actors in London, Henry V1 parts 1,2,and 3, Richard 111, is written in 1592, Titus Andronicus, (written completely in Latin), and at least two, if not three, comedies, The Taming of the Shrew, The Two Gentlemen of Verona, and The Comedy of Errors. This suggests that he moved to London earlier, learnt how to write plays and established himself with the kind of work that forced Marlowe to write Edward 11, just to keep up!
William Shakespeare is one of the best known writers and poets of this century. He is known to every culture and every nation due to his services to not only English literature but also overall literature of the world. Due to his services, he was entitled as the Bard of Avon and the National Poet of England.
Shakespeare was born in Stratford-upon-Avon, England in the year 1564. He was the son of John Shakespeare and Mary Arden. He was raised in the same town having his education at a local school.
Actually it is not exactly known when he moved to London from Stratford-upon-Avon. However, it was after his marriage to Anne Hathaway. The period is considered to be between 1585 and 1592. As after this period, most of his life story is known. He started his own theatrical company in London and was also a successful actor, writer and a producer of stage plays.