How would someone determine who is the author of the works attributed to William Shakespeare?

To determine who is the author of the works attributed to
William Shakespeare, one would need to consult all the available evidence and then try
to make a rational judgment based on probabilities.

Serious
academics who have examined all the available evidence have come to the conclusion that
William Shakespeare is the author of the works attributed to William Shakespeare.  There
is actually quite a bit of contemporary evidence to support this claim, and more of it
turns up from time to time.

In fact, there is so much
evidence (which is easily available; see for instance the literally massive works of
Samuel Schoenbaum) that in order to argue against Shakespeare's authorship, skeptics
have had to deny or explain away or torturously interpret all kinds of hard
data.

If there is a major academic who doubts Shakespeare's
authorship, I'm not aware of that person.  This is a highly unusual situation, since
academics disagree passionately all the time about almost everything -- except this. 
Skeptics about Shakespeare's authorship often have to imagine conspiracies in the 16th
and 17th centuries and then imagine conspiracies in the modern period as
well.

Here's a good site that deals with the issue in some
detail:

http://shakespeareauthorship.com/

The
burden of proof really rests with the people who doubt Shakespeare's authorship.  If we
had even a tenth of the hard, obvious, undeniable evidence in favor of another candidate
that we have in favor of Shakespeare, the debate would be much more lively than it is. 
The debate, instead, now consists of people making claims that seem
dubious.

All I can do is urge anyone who doubts
Shakespeare's authorship to examine the claims of the skeptics very
skeptically.

One of the most powerful tributes to
Shakespeare by a contemporary was written by Ben Jonson, his friendly rival (see link
below), who memorably proclaimed,

readability="6">

. . . the race
Of Shakspeare's mind and
manners brightly shines
In his well turned and true filed
lines

In fact, a very strong
case for Shakespeare's authorship could be built simply from the evidence that survives
from Jonson alone (unless, of course, he was part of some massive secret
conspiracy).