1) Start at
http://dmoz.org/
2) Find a category that you want to edit.
3) Follow the "become an editor" link in the grey bar at the top of the page and fill out the application form in full. If the category you want to edit doesn't have a "become an editor" link, you'll have to find another category, or maybe a subcategory of the one you want.
4) Once you have submitted your application, it will go into the application queue, where it will be reviewed by a volunteer meta editor and your request to become an editor will either be accepted or declined.
You should probably read the [thread=1289]general advice thread[/thread] and the
Editor Guidelines before filling out the application form.