How is As You Like It a subversive play? Consider Shakespeare's use of dramatic methods as well as the context of the play.

In terms of context, at the time that Shakespeare wrote this play, women were not allowed to perform on stage. Therefore, Shakespeare would directly acknowledge the fact that male actors were playing female characters with a number of dramatic conceits that allowed the men playing women to wear men's clothing and pretend to be men in order to fool their fellow characters for various reasons.
In this play, Rosalind must briefly go into exile from the court in the Forest of Arden, and because she is "more than common tall" she decides to dress as a man and calls herself Ganymede. Ganymede is an androgynous figure from mythology, a cupbearer who is also associated with the astrological sign Aquarius, and this mythology contributes to the portrayal of both genders that this character embodies.
Rosalind is in love with Orlando, but when Orlando meets Ganymede in the forest, he doesn't realize it is really Rosalind. Therefore, Rosalind is able to cajole Orlando into telling him all about his difficulties in love. There is also a situation where the shepherdess Phebe falls in love with Ganymede, not knowing that Ganymede is really Rosalind in disguise.
There are different ways in which these situations taken together could be seen as subversive. On one level, Rosalind/Ganymede is effectively portrayed as being bisexual because she must embody male and female characteristics at different times, and must also convincingly fool others when she is in disguise as Ganymede. Although she is not in love with Phebe, she is at first unable to reveal the true reason why she is not available or likely to be attracted to the insistent shepherdess. Phebe's pursuit of Ganymede is somewhat subversive also, in that she immediately turns her attentions from her previous lover Silvius. And Ganymede's attempts to get Orlando to talk about all the ways in which he is in love with Rosalind is somewhat subversive because it is manipulative and dishonest. This also allows for a playful conceit in which the two "men" become friendly and share a bond of affection, which in some productions is played up to allow for an undercurrent of homoeroticism.
The climactic scene of the play which Rosalind admits that, dressed as she is in a way that does not represent her true nature, she cannot satisfy anyone's expectations of her. But when she appears in women's clothing as Rosalind, Orlando realizes who she has been all along, and Phebe realizes her crush is misplaced.