When a theater company or submission opportunity asks for a playwrights’ bio, they are asking for a narrative history of you as a playwright. The most important thing is to note if they have indicated a word length, and to follow that guideline. The theater is likely unable to accommodate in their program or website a longer piece; if you do not follow the guideline your bio will likely be edited to this length and you may or may not be pleased with the cuts.
Often playwrights break their bio into two small sections. The first section is simply a list of their plays (or their top 7-10, if they have several). After that they list theater companies where they have worked. It is not necessary to indicate in your bio which play was performed with which company; this often takes up a lot of your word count.
Playwrights also list if they are a member/fellow at any playwright-specific organizations.
Playwrights then list their education and degrees received, as well as any prominent fellowships or awards they have received recently, especially if any were tied to the play being performed or submitted.
Here is a good template to use when creating a bio:
PLAYWRIGHTS’ NAME’s plays include Title, Title, Title, and Title. PLAYWRIGHTS’ NAME’s work has been produced and/or developed by THEATER NAME, THEATER NAME, and THEATER NAME. Awards include: AWARD NAME, AWARD NAME, and AWARD NAME. Member: ORGANIZATION/THEATER NAME. NAME earned a DEGREE TITLE from SCHOOL NAME.