Firstly, to unlock Director's Cut mode, if you hadn't already guessed, you need to first complete all levels of Story Mode.
Here's some facts about Director's Cut to help you out/incase you're curious.
- Money you have earned/spent on guns/guns you've bought transfer over to Director's Cut, also.
- Unlike Story Mode, you have limited continues when you do, the number you have is 3, rather than somewhat unlimited continues that only half your score each death.
- The levels in Director's Cut are slightly longer due to new little areas/routes you go to being added in, that wasn't in the Story Mode.
The one real tip I would give you to give you an advantage on Director's Cut mode, is using a fully upgraded SMG, due to the amount of bullets a mag has when fully upgraded, and how easily you can spray it at the enemies, just like in story mode, the bosses are the exact same, with the same weaknesses too, so if you remember how to defeat them, that'll come in handy.