Shakespeare is Shakespeare's real name. He was born in Stratford-upon-Avon in 1564, and his father's name was John Shakespeare. He married Anne Hathaway in 1582, and a daughter was born six months later. They also had twins, Judith and Hamnet, who died young, in 1585. Judith lived until 1662.
Virtually nothing is known about Shakespeare's life from 1585-1592. By 1592, however, he is at at least fairly well-known on the London stage. And the rest, as they say, is history.
But, again, yes, Shakespeare is his real name.
(Source: The Cambridge Guide to Literature in English)
Shakespeare is William Shakespeare's real name, but as others have noted, there are variant spellings. We have a good deal of documentation about Shakespeare's life. We know he married at 18, had three children, two of them twins. We have legal documents pertaining to his affairs, including a will. The mystery about Shakespeare comes from two sources. First, we don't have personal letters or journals from him. If he wrote a cache of letters describing his feelings about his plays or his wife or his children or discussed with anyone his thinking behind, say, the composition of King Lear, those letters are all lost or destroyed. So we are left confronting a blank space about the mind and innermost thoughts of England's greatest playwright. Second, if you subscribe to a theory that says only an aristocrat could have written Shakespeare's plays, that creates a difficulty, since Shakespeare was a commoner. This has led to the many elaborate theories about Shakespeare being an earl or a lord using Shakespeare as a pen name. However, those theories have been generally discredited, and the world has embraced the idea that an ordinary man was a great literary genius.
I once heard that his real name was William Webb.
Undoubtedlly, 'Shakespeare' was surname of William Shakespeare. But, with the respect of time, and, with the respect of english Language's development, spelling of Shakespeare has changed.