You should first get the oAuth tokens of the user.
First of you'll need the user's accesstokens. You can retrieve these with
var requestToken = OAuthUtility.GetRequestToken("consumerKey", "consumerSecret", "your return url").Token;
Response.Redirect(OAuthUtility.BuildAuthorizationUri(requestToken, true).AbsoluteUri);
This should ask you to sign in to Twitter. Than you'll return to the return url you've set, with oauth_token and oauth_verifier in the QueryString. These are the accesstokens.
Now you can use these tokens to retrieve the tweets of this user by using
TwitterResponse<TwitterStatusCollection> tweets = TwitterTimeline.UserTimeline(authorization_tokens);
Iterate through these by doing something like this
foreach (var tweet in tweets.ResponseObject)
{
string tweetText = tweet.Text;
}