Warning: Mild Spoilers.
Advisory: you can stack this with the Tin Man and Saw The Elephant achievements to save yourself extra playthroughs.
You'll find that Hard difficulty is rather simple, and that it's simply a matter of taking cover when the time is right, as well as not going guns blazing unless you're properly equipped to deal with the fight.
Tips:
- The only vigors you really need are Shock Jockey, Bucking Bronco, Charge, and Return to Sender. If you upgrade them, heavy enemies become a joke.
- You can skip fights.
- Try to locate Infusions to increase your Health and Salts to max capacity.
- If you're going to use any type of weapon, stay with the Carbine and Machine Gun; when they're maxed out, the MG kills in 4-5 headshots and the Carbine kills in 3 headshots.
Boss Fights:
- Lady Comstock: Grab the crank gun from the tear. Use Return to Sender to deflect her ghost dudes' shots. While you're deflecting, shoot Lady Comstock with the crank gun. When you get enough collected ammo in your RtS ball, shoot it back at the lady and Charge her to death.
- Lady Comstock 2nd Fight: Kill the ghosts and she'll eventually be left by herself. Just hang out outside the bank vault entrance and snipe her to death.
- Lady Comstock 3rd Fight: Go to the little spot where there's a column of stairs on either side. Set up camp and put two Shock Jockey traps on either side of you so that you won't get any nasty surprises. Get a sniper rifle and just shoot Lady Comstock. As long as you stay there and keep the traps up, the ghosts WILL NOT try to come get you. Elizabeth will provide ammo until you finish.
- Engine Defense: Set up a crap-ton of Return to Sender traps on the engine. I mean really, just spam them. If you play your cards right and make sure that the whole thing is covered, then you'll finish the section with only 15-25% health lost. It's really that simple.
Hope it helps! If there's anything wrong, please say so before downvoting. Happy BioShocking!