UploadWithMedia.gs

var API = {
    UPDATE_WITH_MEDIA : "https://api.twitter.com/1.1/statuses/update_with_media.json"
};

function oAuthConfig() {
    var oAuthConfig = UrlFetchApp.addOAuthService("twitter");

    oAuthConfig.setAccessTokenUrl("https://api.twitter.com/oauth/access_token");
    oAuthConfig.setRequestTokenUrl("https://api.twitter.com/oauth/request_token");
    oAuthConfig.setAuthorizationUrl("https://api.twitter.com/oauth/authorize");

    oAuthConfig.setConsumerKey(ScriptProperties.getProperty("consumer_key"));
    oAuthConfig.setConsumerSecret(ScriptProperties.getProperty("consumer_secret"));
}

function postImage(tweetText, imageUrl) {
    oAuthConfig();

    var boundary = "cuthere",
        status   = tweetText,
        picture  = UrlFetchApp.fetch(imageUrl).getBlob().setContentTypeFromExtension(),
        requestBody, options, request;

    requestBody = Utilities.newBlob("--" + boundary + "\r\n" +
        "Content-Disposition: form-data; name=\"status\"\r\n\r\n" + status + "\r\n" +
        "--" + boundary + "\r\n" +
        "Content-Disposition: form-data; name=\"media[]\"; filename=\"" + picture.getName() + "\"\r\n" +
        "Content-Type: " + picture.getContentType() + "\r\n\r\n"
    ).getBytes();

    requestBody = requestBody.concat(picture.getBytes());
    requestBody = requestBody.concat(Utilities.newBlob("\r\n--" + boundary + "--\r\n").getBytes());

    options = {
        oAuthServiceName : "twitter",
        oAuthUseToken    : "always",
        method           : "POST",
        contentType      : "multipart/form-data; boundary=" + boundary,
        payload          : requestBody
    };

    return UrlFetchApp.fetch(API.UPDATE_WITH_MEDIA, options);
}

function tweetTest() {
    var tweetText = "GAS update_with_media テスト",
        imageUrl  = "http://blog-imgs-38.FC2.com/n/e/n/nenesoku/120ebf562bdd2446fbf8459b3e6265e7.jpg";

    postImage(tweetText, imageUrl);
}

UrlFetchApp is only available on server