I am the developer of Tweetinvi which is Portable Class Library allowing you to access Twitter REST API very simply.
Would I just need the twitter API and streamwriter? Is there anything I am overlooking?
In theory it would be the case but there are many things you need to consider when accessing the Twitter API. Authentication, not documented endpoints, not documented parameters, fields that have different type of values based on the enpoint (e.g. string vs. int)... Tweetinvi will take care of this for you and in addition you will have a access to a set of tools simplifying your life (e.g. RateLimits).
By using a wrapper you won't have to consider any of the problems you can encounter from the Twitter API.
Here is an example of what you want to achieve:
Auth.SetUserCredentials("CONSUMER_KEY", "CONSUMER_SECRET", "ACCESS_TOKEN", "ACCESS_TOKEN_SECRET");
var userId = 1577389800;
var userLatestTweets = Timeline.GetUserTimeline(userId);