You can redirect your URL to your WP install, but that's not what you want really.
Th egyst here is that wordpress.com and wordpress.org are not linked in any way. They are 2 different animals.
Here's my suggestion. First of all, get wordpress installed on your hosting, with your new domain pointing to it. Don't worry about your content yet.
Many hosts offer 1 click WP installs, see if yours does. If not (or if you want to start learning how to do things manually - which I highly recommend) here are the install instructions -- http://codex.wordpress.org/Installing_WordPress
Notice where I got that help document from - it's the WP manual basically, The Codex. I recommend you get very familiar with it, it can help you with many things.
So now you have a brand new WP install on your host with your domain. You can move your content now, but I wouldn't do that yet. I'd get things looking like you want them. Maybe consider a theme. Unlike over on .COM where you just pick a theme, on .ORG you must find and install a theme. You can buy all sorts of themes, but I'd recommend starting your search here: http://wordpress.org/themes/ If you don't find what you like then you can broaden your search.
Next up is functionality. Not every feature you had on .COM will exist on .ORG out of the box. We have plugnis to add functionality http://wordpress.org/plugins/ Maybe you don't need any to start with, don't add them without thinking about what functionality you need. Too many plugins can slow your site down. You can always add them down the road, later on when you need new functionality.
So now we should have a wordpress.ORG self hosted installation, we've considered appearance, we've considered functionality, and we know where the help documents are.
It's time for content - you can import everything from your .COM installation
You would need to export everything from your .COM and then import to your new install.
After everything is set up, maybe make a post at your old site telloing readers to come here.
Hopefully this is a good amount of info to get you started!