Just play through all ten chapters successfully. It'll unlock either during the ending credits or once you're back on the Main Menu. Two difficulties are locked, but they do stack, so, initially, you'll only have access to Normal and Hard. After completion of either, you'll unlock access to Expert and Impossible. As the difficulties are not that challenging, other than when going for Mega Star trophy (50 stars, 5 star'ing all ten chapters), I just completed Hard and then Impossible to pop all the difficulty-related trophies.
For that profile you're playing on after beating the game once, you can just do chapter select and choose the difficulty; after each level on Impossible, I'd back out to my profile and pick the next chapter, allowing me to choose a gun from the collection--which helped in obtaining a few 5 stars I didn't get on my Hard playthrough. Once your weapons are upgraded, the game gets even more manageable!