If you have already signed up to the 'First Access Club' through either pre-ordering Duke Nukem Forever or through the Borderland GOTY Edition you should have already received the email from Gearbox announcing that you can finally get your hands on the demo on June the 3rd. There's no date on when everyone else can play the demo so if you want to put Duke through his paces before the full version of the game is out its through the pre-order or pick up Borderlands GOTY.
Members of the club should also have received emails getting you to reconfirm the platform you want to get the demo for and if you haven't now you know how soon you have to confirm it.
Duke Nukem Forever is slated to hit the US on the 14th of June and everywhere else on the 10th. The official website can be found here.