There are a few different ways to do that. The best way, and the only way really available to you right now is to answer threads on the forums. In addition, people will occasionally enter the chat room with questions that could help with.
After a month of being registered at TeenHelp, and provided you have at least twenty-five posts in the forums, you're qualified to volunteer for a staff position. There are three main staff positions to apply for (at first)
Buddies (Like myself) welcome new members and make sure people are able to use the site's features. Buddies hang out in chat a lot, send Visitor Messages to new members, welcome members in "Arrivals and Departures" and just generally present a friendly and welcoming face to new and old users alike.
HelpLINK Mentors give advice and support through the email based HelpLINK system. I'm not one, so if you wanted to get a better idea of what the job entails, I would talk to a HelpLINK mentor (their usernames show up in pink on the site) to get an idea of what the job is like. There's also a description of the position on the application form (as there are on all of them)
Live Help Operators give advice and support through the Live Help system (similar to an instant messaging system) Live Help Operators names are in a sort of dark, tealish blue-green (I'm not sure what to call the color, to be honest
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Applying is pretty simple, you just fill out the form for whatever position you want, and then you submit it and wait. Be patient! Sometimes you won't hear back for close to a month. In addition, don't get discouraged if you get denied the position the first time. You'll be able to try again in another month, and oftentimes second time applicants have a better chance of getting chosen anyway.
Hope this helps, I know it was kind of long! Tell me if you still have any questions.