It looks like you partly installed WP in a subfolder called WordPress (note the capitalization).
When I go to your domain, it acts as if I'm logged in as you and that is caused by the index.html file.
You should do the following:
1) Delete the index.html file
2) Copy (do not move) the index.php and .htaccess files from the WordPress folder into the main folder of your site.
3) Open your root directory's index.php file in a text editor
4) Change the following and save the file. Change the line that says:
to the following, using your directory name for the WordPress core files:
5) Login to your site. It should still be http://example.com/WordPress/wp-admin/
6) Go to Settings -> Permalinks and press save (that will re-set your permalinks)
That should be it If that doesn't get it fully working, let us know, or ask one of our awesome support folks for a hand.