Bart doesn't have many lines but he's an interesting figure. We only ever see him in character in the Induction (he dresses up as a woman and pretends to be Sly's wife as part of an elaborate practical joke). The thing is, Bart is really good at acting. With a little direction from the Lord, Bart manages to convince Sly that he is an obedient wife who is ready to serve Sly's every whim. He's a little too convincing, in fact, because Sly is so convinced and turned on that he expects his "wife" to get into bed with him.
Bart's role in the play alludes to the practice of using boy actors to play the roles of young women on the all-male Elizabethan stage. They were considered attractive or "pretty" enough to do the job and their voices were appealingly high-pitched because they had not yet hit puberty. In other plays such as As You Like It and Twelfth Night, Shakespeare more fully explores the homoeroticism of Elizabethan theater. Here, however, the issue is not as fully developed.
Taming of the Shrew certainly raises the issue of the exploitation of boy actors. Bart is never really given a choice as to whether or not he wants to perform and the role is a bit dangerous given Sly's behavior. When the Lord sends one of his men to fetch Bart, the nobleman relies on the fact that Bart is his servant and owes him his allegiance and "love":
Tell him from me, as he will win my love,
He bear himself with honorable action,
Such as he hath observed in noble ladies
Unto their lords, by them accomplishèd: (Induction.1.114-117)
Bart's convincing performance also does much to call into question the performative nature of Katherine's final speech. If being an obedient wife is so easy that even a male child can do it, we wonder if Kate isn't playing a convincing role as well.