This is pretty self explanatory. After beating the game on Normal or above (Easy doesn't count!), you unlock new game plus.
Starting new game plus removes counter alerts, makes enemies tougher and faster with more weapons/shields/armor/stunrods than normal, boss fights will require more effort
and sneaking segments see more armored thugs.
Starting a new game plus does not affect the difficulty of Riddler's revenge or the Riddler's campaigns, so don't worry about that when going for Perfect Knight 2.0. For a sneak preview of what fights and sneaks are like, you can go into custom challenges and turn on the scattered weapons, increased aggression, one hand tied (combat), and decreased health modifiers.
Boss strategies (cheap tricks) in no particular order
For other strategies, you can check the xbox 360 version's guide by searching this achievement in the search site bar in the upper right.