Though there are a few other archives around, the main place to post and share Newsies fanfiction is Fanfiction.net, so you'll probably want to know how to get your work online there.
First, before you post, go back and reread through the writings basics section. Make sure you're doing all of those things, with special emphasis on paragraphing and punctuation. You are? Good.
Next comes beta reading. Find a beta reader, and treat her well. Bake her cookies. Offer to have her children. Whatever. But do it after he or she finishes with your story, because you want an accurate, unbiased opinion. Ideally, a beta-reader is someone you're friendly with, but not so close to that s/he'll feel bad telling you what's wrong with your story. Ask your beta to please tell you if there's any spelling/grammar/punctuation mistakes, if there are any word choices that feel too awkward, any gaping continuity errors, etc. Furthermore, if there's a scene you're unsure of (especially if it's a pivotal scene, like a climax) ask your beta for feedback. Did it work? Why not? What could make it better?
Then, and this is key, listen to your beta. If your beta says you went horribly wrong in the third chapter, fix it up. Ask her to read it again. Improve it. And then, this is even more important, thank your beta-reader. A good beta reader is the best thing you can do for your fanfic. Trust me, Cigarettes and His Mother's Eyes (my first major attempt in the fandom) would not have happened without my betas, and even had it, it would have been crap. Find a beta-reader or two, and treat them well. You'll be very glad you did.
Spellcheck your story. Yes, I'm saying it again. Do it, or face public embarrassment. Even if no one catches you on your bad spellingyou'll know, somewhere inside of you. You'll feel dirty and wrong. Save yourself the trouble. Spellcheck now, before it's too late. I can not stress that enough.