I think the easiest way is to pay $49 to submitedge.com and they will do this for you. Details are on this page – submitedge.com/dmoz_listing.html
Otherwise if you want to do it for free, just follow the suggestions that DMOZ give on their website. You have more of a chance of being accepted if your website is somehow unique, or offers something unique to the other listings already present.
One of my sites actually got accepted into the DMOZ directory in 2008. It stayed there for about 2 years before it was suddenly removed for no reason. Being in DMOZ isn’t necessarily permanent, and it doesn’t guarantee high Google rankings. I noticed no significant rank changes after I was accepted in DMOZ. Just wanted to point that out before you apply. Best of luck 