Hard mode on X-Men Origins: Wolverine is not much more difficult than normal mode. The enemies do more damage and you do less damage. So the mantra for finishing hard mode is a lot of patience. Follow the smart road: when coming across spawning machete guys, kill the leader first; when you can throw enemies off high ground or impale them on a forklift, opt for that instead of trying to kill them the conventional way; use the quick kill feature often, especially against Ghosts and Shifters; take down machine gunners before going after melee enemies; be patient when fighting Levathian and W.E.N.D.I.G.0 Prototype.
Also, make use of your Mutagens and Skill Points. Equip the right Mutagens to gain experience and reflexes faster or opt for faster healing or less damage. With Skill Points, buy more health and upgrade the damage you do with your claws. If you use Berserk mode or Claw Spin a lot, make sure to upgrade these as well. Gather as many dog tags and health bonuses as you can find. The extra experience from the dog tags can bring you extra Skill Points and a bigger health bar means less dying.