Beating the game on any difficulty unlocks the Hard Core setting. The difficulties DO NOT stack in this game so beating it on Hard Core does not unlock Mission Impossible.
Weapons: I recommend buying only the force gun and using it in conjunction with the Plasma Cutter. Having more than these weapons in your inventory or safe will cause ammo to drop for weapons that you do not use and will spread your ammo very thin for the guns that you do use. The Bloody Force Gun that comes with the Martial Law DLC will allow you to get this gun at the first store for free as opposed to waiting until Chapter 7 for the schematic.
Armor: This is much easier if you have bought the CE or Martial Law DLC Pack as you will get a 15 slot, 15% armor suit at the start of the game making inventory space and damage control more manageable. The Bloody Vintage Suit that comes with the Martial Law DLC gives you an additional 5% damage with the Force Gun. NOTE: You can equip any suit to get the extra inventory slots and armor, and then revert to another suit and still keep the # of slots and the armor bonus of the best suit.
Inventory: Keep 4-5 medium med packs. The large med packs are extremely expensive at 10,000 credits. The medium med packs are half the price and do the trick, while allowing you to keep your credits for the more essential items you will need. Keep 3 stacks of stasis kits (9 kits) for getting yourself out of tricky situations. For ammo, I recommend 2-3 stacks of plasma cutter ammo as it is fairly expensive and you will only need it in precision situations (i.e. Hangar Boss in Ch. 1, Brutes, and final boss) where hitting a particular part of a necromorph really matters. 5 Stacks of Force Gun ammo should leave you with plenty of wiggle room until you get to the next store. Plus, force gun ammo is cheap! Use any spare money you may have after restocking to buy power nodes to keep your equipment upgraded.
Saves: I recommend using your saves in the later chapters and utilize the disc switch glitch as your first save, but remember you can't shut your game off until you actually use a save. If you are having trouble getting to Chapter 7 without saving you can use it early, but I wouldn't recommend using it any sooner than chapter 4. If you make it to the disc switch without using a save, then I recommend saving on Chapters 10 before the hallway with the two Brutes, Chapter 11 (before entering the 'Exploration Mine' and the Drill Ride) and 13 after defeating the final Brute. For your Chapter 13 save, take the Elevator up to watch the cutscene of the Marker, then ride back down and save so if you die you don't have to rewatch it.
Here are some additional threads with tips from other members that may prove useful.
Hardcore Tips by SquaredSumo705
Hardcore Tips by Jumnious
Hardcore Tips by DreamDevice