Don't over explain and describe his actions, thoughts, motivations, etc.
First, try to establish that he is, for the most part, not a happy camper. To what extent is purely up to you. If he's suicide revering, to go out and say it full blown head on isn't the best method. Make sure important, grand, thoughts/feelings like that are relevant or brought up in the right context.
Then throughout the story, if his actions are more or less motivated by his 'emo-ness', don't let him, or someone else explain/describe it. Try to keep it neutral most of the time. Two ways we'll get the picture of what validates his actions/thoughts are:
Every action has an immediate connotation and a direct, unmistakable root. Doing cartwheels is a generally positive action, and you don't really need to explain the person is a happy-go-lucky type of person to validate it because they got a pay raise or for no reason at all. It's pretty much implied. You can explain/describe if you want, but you don't have to do it ALL the time.
Second, every action has a reaction. After your character does something, even if you explain it neutrally, the reaction/consequence from that action that occurs sooner or later will then explain his prior actions/thoughts. Now you don't have to explain everything descriptively, or like an observation made by himself or the writer. The only time observation that directly point out his character/personality should be voiced/thought are from other people. Examples would be his parents are worried about him, or he's the type of person who people shy away from during recess, or school faculty treat him with a little more fondness than regular students because they want to make him feel like he's not abnormal. These examples are probably out of context to your story, but they're just examples.
A plus for doing it this way is that when the reader is reading his story, it makes the reader feel like the character thinks his actions are already validated, or normal (to him, that is), and feel like we're moving along with the character, not the narrator or writer.
These tips just make it more bearable, I believe.
If you want the weight of his emo-ness to make us scream at him, or feel bad for him, then apply a few drops of strong emo-ness here and there. Also make crucial moments very crucial. The absence of conviction can obliterate crucial moments.
I fear what I'm about to say isn't too radical to comprehend:
A reader can more easily bear a story of nothing with excellent story telling, as opposed to a story chalk full of flowery writing, vivid descriptions, or tangible characters, that allude to the same nothing.
Trust me, readers greatly appreciate story telling.
I hope this has helped