Were there any prerequisites to become a playwright in 1589?

There were no hard-and-fast requirements or prerequisites
for becoming a playwright in England in 1589, partly because professional drama and
permanent theaters were relatively recent innovations. However, the following
qualifications are some of the traits that would have helped someone establish a
successful career as a playwright at that
time:

• a decent education.  Many playwrights were
  expected to be well-read, especially in the literature, history, and mythology of Greece
  and Rome as well as in the literature and history of continental Europe, especially of
  France, Spain, and Italy.

A university
education was not a prerequisite (two of the greatest dramatists of the age – William
Shakespeare and Ben Jonson – did not attend universities). Nevertheless, a solid
grounding in the kind of texts just mentioned could be achieved in Elizabethan grammar
schools and through independent study. Many dramatists (including the so-called
“University Wits”) did have college educations, but such educations were not absolutely
necessary.

• Some familiarity with English
  dramatic traditions of the distant and especially the recent pasts. A talented dramatist
  would have known something about the so-called “morality” and “mystery” plays of
  medieval England, but he (since dramatists were typically male) would especially have
  been expected to know about the kinds of plays that were popular and financially
  successful at the time he was writing. The London theatrical scene was highly
  competitive, and the best playwrights were those who could meet the tastes of the
  time.


• A skill for writing certain
  kinds of plays. Some writers, for instance, were especially skilled
  at writing comedies; some were especially skilled at writing tragedies; some were
  especially skilled at writing histories, and so on.  Some of the most talented
  playwrights (such as Shakespeare) could write in various genres with comparable success.
  

One recalls Polonius's reference, in
Hamlet, to

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The best actors in the world, either for
tragedy,
comedy, history, pastoral,
pastoral-comical,
historical-pastoral, tragical-historical,
tragical-
comical-historical-pastoral . . .
.

• A
  talent for writing for different kinds of actors. Some plays were written to be acted
  entirely by boys; some were written to be acted by adult men (with boys playing female
  roles).  Sometimes playwrights were especially well known for writing
  either for the adult troupes or for the boy
  troupes.


• A willingness and ability to write in
  collaboration with others.  Many plays of the time were the products of multiple
  hands.


• Some practical experience in acting was an asset.
  Shakespeare was an actor; Jonson acted; various other playwrights began as actors before
  becoming writers, while some remained actors while also
  writing.


• A strong familiarity with the generally accepted
  moral and religious values of the time. Elizabethan playwrights could not, for instance,
  openly advocate atheism or call for an overthrow of the government. Plays had to be
  approved, before performance, by a government censor, and playwrights had to know what
  would or would not pass the censor’s scrutiny.  It made little sense to write a play if
  the play was so thoroughly unconventional or radical that it could not be performed.