There’s primarily two components you’ll need to write. Since all requests must be authenticated, you’ll need to write code to perform OAuth authentication. Twitter has documentation here, which also includes steps to build the OAuth header:
Next, you’ll need to build the code that makes the HTTP request. You can use HttpWebRequest or HttpClient. I saw today where MS released a new beta of HttpClient for portable libraries. You’ll build your request and add the OAuth header, pointed to the Twitter API endpoint for status updates at https://dev.twitter.com/docs/api/1.1/post/statuses/update. You can do a search for “HttpWebRequest Twitter” and get plenty of examples - the exception being that many of the blog posts are old and username/password authentication has been deprecated a while ago and replaced with OAuth.
I know you would rather not use a 3rd party library, but if you decide you don’t want to write all this code and change your mind, I wrote a 3rd party library named LINQ to Twitter (http://linqtotwitter.codeplex.com/) that does support Twitter API v1.1.