Here's a little guide on how to unlock the New Game + in Dark Souls.
New Game+ Mode: To get new game mode + mode, you must first successfully complete the game. Then you can select the New Game + mode form the main menu and you will get to begin a second playthrough with all of your previously obtained souls, weapons and armor.
Keep in mind you don't get to keep unique items like keys or embers, and the enemies will be a lot more difficult during the second playthrough. The enemy health and their attack values are raised by 2x as much as default.