You sign up by submitting your app packaged in a PPA before the deadline (9th July 2012) through the MyApps portal, along with the App Showdown Participation details text file to indicate you want to take part in the contest.
That's step 4 in the Ubuntu App Showdown page.
Sending a message to @ubuntuappdev on Twitter is not exactly necessary but is highly recommended to let the judges and others know how your app is progressing.
This section of the App Showdown details page makes it pretty clear:
Spread the word
Keep everyone updated by blogging about your app’s progress and post
the blog entry link on Twitter with the #ubuntuappshowdown hash tag.
Be sure to send your blog entries to the UbuntuAppShowdown reddit or
as a mention to @ubuntuappdev – the judges like to see your progress!
The blogging and tweeting is just a means of spreading the word about the App Showdown and your app.