First of all let me just say in advance that I have no experience with playing MonsterHearts. It is in my "to-play-list" mainly due to this little section in the rules, but all of my mechanical aspect of the answer is based around reading alone.
2 quick notes before I dive in: Firstly, it is better to decide on one aspect of the "LGBT-Experience" to focus on. Secondly, I believe that after we'll know what characters have been chosen it will be much easier to suggest things.
Part 1: Skins as metaphors
I truly believe that at least some of the Skins are quite straightforward in terms of queer metaphors, but some of them are a little bit harder to pull. In this part of the answer I'm gonna move from Skin to Skin and write how I see them in terms of queer metaphor.
The Chosen
The Chosen is a person who is complete with his/her sexuality. It is a person who had the courage, time, company, help or else to find in herself those things and to accept them. A Chosen one is healed by Sex, and why? Because she knows that it helps her to fill herself, because it shows her again and again that she accepted it and that through having sex with others, she can show them that there's nothing there to fear, no matter their sexual attraction.
But she believes that she has to go this path alone, because her friends are either hetero or still in the closet. And she know that she can't out them (right?). More than that, though, the Chosen sees herself as defendant of her kind, defending those who are being bullied because of their sexual attraction or because of their gender identity.
The Fae
The Fae more lives as one of the fears of the LGBT community. It is a person who outs other. He asks them to promise him to come out of their closet, whether they are ready for it or not. And he can pretty much make sure that if someone doesn't come out by himself, the Fae will do the work "for" him.
The Ghost
The Ghost is the shy person who lives in her closet and tries to disappear so no one will notice her. She knows that when they'll see her they'll know, and then she's lost or doomed or even both. It is the person who wants to be accepted by society but who doesn't believe that they'll accept her.
All of this, though, comes from a trauma in the past, probably an outing by another or a coming-out move from herself, that resulted with her being shunned or ridiculed or both. She carries this trauma 'till this day, fearing for it ever coming again.
The Ghoul
The Ghoul is a person who lives through her desires. She lives to have sex with other, and maybe because this is the only thing that she feels. This is actually the stigma that is being put on Bisexuals and Pansexuals, so this is quite an easy thing to pull as a metaphor.
The Infernal
This one is actually quite hard. I will go here with a metaphor to a certain kind of transsexual (well, not a kind, but you know…). A transsexual who felt so bad about her body, and couldn't wait anymore for the operation that she made a deal by herself. She bargained a part of herself for the money, maybe promising her new body or soul in return. She knows that one day she will be asked to pay back. Her new and right body is not a gift, and it didn't make her happy as she thought it will, but she will change it one day. One day she will pay her debt and the transformation will be complete.
This brings her to the metaphor of a transsexual caught in the middle of the transformation, which is hard to play just like it sounds, with all of those conflicting impulses and the fear for it being stopped in the middle or something.
The Mortal
The mortal is one of the more famous "types" of the Genderqueer, it is a person who sees the gender as mere game and nothing more. There are neither male nor female, it is just a game, just a show that who pull, an act that we play. By playing this game in terms of sexuality also, the Mortal here tries to show to others the truth about gender and gender roles. ,About the true nature of our social codes. They'll have hard time getting it at first, not to mention accepting it, but one day they will. And when they'll see it the same way, it will be much much different. We will look in each other eyes and they will thank me and love me. And I may love them back, or just continue to show the truth to others.
The Queen
The Queen can be one of 2 main ideas, as I see it. The first one can be that she is one of those who try to "cure" people from their "LGBT-Disease". She believes that by showing them what true sex is, and what true attraction really is, she can "cure" them from their "situation". It is, to a certain extent, the dream of the most conservatives of the conservatives. She will "save" their souls from the hell that they are headed to.
Or she can be quite the opposite, as another "kind" of the Bisexual or Pansexual. She tries to show to all of them that deep inside, we're all Pansexuals. Freud said it back in his day and he was right, so why the hell did we forget this? We're all just pawns to our attractions and wants and needs. When we'll finally acknowledge it everything will be much different, much better.
The Vampire
The Vampire is quite hard to figure from another reason- one can look at him/her and find whatever one wants. We can look at the Vampire as a metaphor for homosexuality or for heterosexuality and how great it is, we can look at it as a metaphor for AIDS or as a metaphor for Bisexual.
I still can present here something, and this is the metaphor of the Transsexual after the Transformation. Even after the Transformation one has to continue to take hormones in order to stay in the new body. More than that, people wanna become vampires in order to stay beautiful, and what can it be if not to be one with my true body? My perfect body? And here you've got your metaphor, you drink blood (hormones) in order to keep the body (the right one, the one you sacrificed so much to get) intact and perfect.
The Werewolf
For me, the Werewolf was always the metaphor for the person who doesn't control her body, to the one who got the body that she doesn't deserve. The Werewolf can, thus, be a metaphor for the pre-Transformation Transsexual. More than that, having to keep the transformation secret can be a metaphor for the "conceal-it" attitude. Another nice touch can be that the Werewolf is actually a girl trying to masquerade as a boy, and her three days of full moon…
The Witch
The Witch has always been a metaphor for Lesbianism and I don't see why it should change.
Part 2: General suggestions
But those are just metaphors, and without a game that surrounds them they don't count for that much. So here are some general suggestions and tips that I hope will be useful for you.
Read a lot
This is a very sensitive subject, and as such should be taken seriously. When you tell stories about Transsexuals it is pretty clear to most of us when this story rings true to the subject and when not. This means that you'll have to learn a little bit about the themes of your campaign and a little bit about queer theory. There are some wonderful Gay and Lesbian Readers about the subject, and reading parts of them can prove quite useful. There are also some Internet blogs created by the LGBT community for the LGBT community. Some of them have things aimed at those who try to find themselves. Reading those can also prove to be quite useful. Look for coming out stories, as they will probably prove the most useful for you in the short term, regarding this type of campaign.
Watch a lot
More than that, it is really useful to see how LGBT people are represented in the media. It can show you from what to keep a distance and to what you should strive. There are 2 movies that jump to my mind that I would start from them as they present a huge array of possibilities and ideas, in addition to showing some of the "fights" for recognition and to find and accept oneself. Those are: La Ley Del Deseo and Ma Vie en Rose. Both of them tackle Transsexuality, with the second revolves around the search for parental approval and the first about the day-to-day lives of LGBT (it revolves mainly around a gay love triangle but also tackles Transsexuality and love between parents and kids). More than that, going through the winners of the Teddy and GLAAD prizes can prove invaluable.
Present NPCs from all of the spectrums
It can prove really hard to do so but it is still invaluable. You can present them with metaphors or without them but they should be here. The characters and the players should both see that their not alone and that other people are still quite different. There are many "types" of Lesbianism, from the Butch to the more Femme, and showing the spectrum will help you to rise beyond those stereotypes. Think for example how the Witches in Buffy are different from one another, or Naomi and Emily in Skins. Look at the difference between Ludo and Tina in the movies I presented a little bit before. The fact that they are Transsexuals is not the only defining trait that they have.
Look for their struggles
LGBT people are like teenagers in that they search for themselves, trying to define themselves all the time. They try to rise beyond the stereotypes or to hide beneath them; they look for approval or fear from the disapproval and so on. More than that, the moments of discovery can be quite shocking. The moment when a religious person discovers that he's gay, for example, has been the subject for a huge amount of movies and novels.
But their struggles can also be after they've been identified, consisting of the stigmas bestowed upon them, or the cursed from the religious parties. Maybe their teachers will present them "cures" for their conditions, or the hormones will stop coming.
God is punishing me
One of the most common tropes of LGBT stories is the arrival of AIDS to one's surroundings. It doesn't have to attack only LGBT people (and it can even grant an Oscar prize…) but the main deal here is how it is being understood and looked upon. Do they look at it as a punishment? As an empowerment? Something else?
And an end
As this answer came to this length, I think that I'll just stop here. There are many things more to get from here, may more areas to explore. Good luck, and I'll be glad to hear how it went.