Microfilms of old newspapers at the library will unearth birth announcements, complete with parent’s names, date of birth, and other information that might be required. Should your request be questioned, there are many logical reasons why one might seek a birth certificate for a person other than himself. A distant relative doing genealogical research, perhaps?
If you've adopted a moniker that is common (but not too common), you might consider co-opting the birth certificate of a living person with the same name. Since there are a great many Robert S. Johnsons out there, is any harm done if a particular Robert S. Johnson born in Dayton Ohio on 7/7/77 now has a doppelganger? No... unless you're attempting to pass yourself off as Robert S. Johnson in furtherance of criminal activity, attempting to receive any financial benefit, or using it to identify yourself to a cop. You simply wish to be known as Bob Johnson and in support of this you are fobbing off a birth certificate issued to one. Find this prospect ethically unsettling? Then by all means don't do it.
Another way to engineer a Virtual Self is to check the obituaries in on-line editions of local newspapers until you stumble upon a person of your approximate age who had an agreeably generic name. The obituary will list his birthplace, enabling you to contact the Bureau of Vital Statistics in that state to order a certified copy of his birth certificate. (But act quickly; once the death certificate has made its way through the bureaucracy, the birth certificate will be forever stamped "deceased".) When all is said and done, you'll have a certified, bona fide, legally-obtained document with "your" name on it that might come in handy when government-minted ID is required. ("What? You need I.D.? From me? But I lost my wallet yesterday! Look, I'll come back with my freaking birth certificate. It doesn't get more real than that, right?")