At age 13, the big thing for people is usually early puberty. They're sprouting body hair, their voice is possibly changing, they're putting up with a testosterone-fueled sex-drive for the first time. And for a trans person, it feels like their body is betraying them.
At that age, my main feeling was that my body was getting less and less like I wished it could be every single day, and I was looking over at the girls who were starting to fill out and become beautiful, and who didn't have to put up with the body hair and the bulky male muscles and the voice changing and all of that, and I was so freaking jealous.
Also, at that age, you can take any number of routes. Some trans people at that age are visibly effeminate, and it's probably the age where they're facing the most social ostracization, because it's the age where most boys "man up" and start calling people "gay" all the time, and teasing, and just being downright nasty to anyone that they view as an inferior male specimen to somehow prove their own manliness. Such a person is probably very visibly effeminate, is probably teased about being "gay" constantly, probably beaten up often, probably treated like dirt by his male peers. Also, some trans people at that age, like me, have no idea what's happening to them. They weren't too gender-nonconforming as kids, and yet now all of a sudden they find themselves envying girls and hating their own male puberty and wishing that they could be female. They've never felt this before. It's so confusing. In a society that's constantly trying to tell them that maleness is superior, that they should be toughening up and manning up, they appreciate niceness and softness and sweetness, and just don't understand why their male peers are acting like ->-bleeped-<-s all of a sudden while suddenly they're wishing they were girls. It's confusing. It's scary. You feel all alone. You wonder what's wrong with you. You feel like you can't tell anyone about it because you'd be a freak if you do. Your dressing in female clothes is probably in private, because you're so afraid of being seen as a freak. Some in that situation just become disenfranchised and reclusive and depressed, ashamed of themselves. Some go into denial, trying to man up and be the perfect guy, never letting on to their trans desires.
It really depends on the character. It's such an individual thing. There are so many different ways that different personalities react to being trans. And people figure it out at different ages, in different family structures that lead them to be either more open or more ashamed about it, it affects people in different ways, and manifests itself in different ways.