First you're going to need to download some Club Penguin media. You can get it all from http://icer.ink.
Your VPS is going to need to have a webserver (I reccomend you use nginx), PHP and a MySQL database installed.
Upload the media to your webserver webroot and extract it, then configure subdomains (respective to the ones from icerink) on your own domain and your webserver to point those domains to the respective webroot folder in the media you downloaded from icer.ink.
Next you'll need to crack the Club Penguin client to disable domain locking, this essentially just involves replacing actionscript constants within files inside the /play/v2/client/ & /play/v2/games folders inside the game media directories. You can do this manually using JPEX free flash decompiler or you can use a tool I wrote in Python specifically for this process. You'll also need to modify global_crumbs.swf to point the game client to the correct address and ports that Kitsune will run on.
Edit the files inside your play subdomain webroot to work on your domain (instead of clubpenguin.com), or just create a new HTML webpage which embeds club_penguin.swf.
Now go and clone Kitsune out of its official repository, and configure it for your server.
Execute Kitsune.sql on your database.
Create a user account in your Kitsune database, run Kitsune on your server, and login via the play page.
Very brief, and I purposely missed out huge amounts of detail but you can always ask for more specific help.