Be aware that on Hard difficulty, there are no checkpoints like in the other difficulty. The only time a "checkpoint" type of save is made during a level, is when you reach the usual level midpoint, and you see a new "area name" up on top of the screen. When you reach this point in a level, if you then die, you'll start from that point, rather than the beginning of the level.
That being said, play the game with the intention of never dying.
The key to succeeding, is to master the art of pouncing, dodging, and countering.
Pouncing is less important in this game than it is in the first game, but can still be useful when you want to finish an enemy pretty easily, since normal sized enemies cannot block a pounce. Simply pounce on them, and perform a 2 or 3x Square, followed by Triangle combo.
When an enemy flashes briefly just before an attack (also an "!" icon appears above his head), this is the signal for you to perform a counter by pressing R2 when that happens. Counters are usually 1 hit kills, except for some of the big or fat enemies. Despite that, you can usually counter back to back, so memorizing the different enemies' attack patterns really becomes helpful in this difficulty.
Don't ever try to pounce a big or fat guy. You should be able to grab them no problem.
My weapon loadout was always Shank, Hammer (heavy), and Shotgun (firearm). If you stuck to your combos pretty well (ie - Grab with R2, then Square, Square, Triangle, and then shotgun the body as its flung from you, or laying on the ground, for usually an instant kill after this combo), you should be able to get a rhythm going with enemies.
When you encounter shielded enemies, Hit them with Triangle (1 hit for normal sized enemies, about 3 Triangle hits for big enemies) to destroy the shield so you can follow up with a real combo.
There are 8 levels in this game, and at the end of each one is a boss. Usually in the boss fights, you have to learn the boss' patterns of attack, so you can dodge appropriately. Then they usually require you to hit them with some combos until an R2 (usually it's R2) icon appears over there head, which is a trigger for a cinematic stylish combo attack which does a good amount of damage to the boss. 4 or 5 of those R2 combos and the bosses should die. Might even take less depending on how much you perform regular combos before dizzying the boss.
Let me know if these tips are useful.