This mode is unlocked after beating the game on any difficulty. You initially have only have 1 minute to finish each chapter, but kills earn you more time. The timer will stop during cut scenes and slows in bullet time. Headshots, cover, precision and bullet time will be crucial on this playthrough as well as speed. Use the Painkiller Location Guide. Painkillers are very handy in your NYM:HC playthrough as well as Old School and Hardcore playthroughs.
If you die in this playthrough you will restart back at chapter one, so be very careful when toward the end.
Also see Payne In The Ass achievement description for Harcore playthrough tips.