I am trying to post a single tweet in my Windows phone application to any twitter account. I don’t want to complicate my code using external libs or APIs. If they are unavoidable, I will include. I don’t need to read any tweets or need to persist my connection. Its simple, The user supplies a username, password and a tweet message and hits tweet.
The famous temple run in iPhone has exactly what I need (screenshot below).
Please guide me in C#

This is possible via XAuth. The technique is not straight forward though, your request will have to go through the Twitter API team and once they approve they will give the xAuth Access. See this other SOF answer for more details.