As the others have said, you need to complete EVERY chapter on hard difficulty.
I too recommend playing through the game on easy difficulty first, as to be honest, the jump up in difficulty isn't that large. The main differences are:
You will NOT see any red dots on the map to indicate where enemies are, so you will need to be sure an area is clear of enemies before you move on, otherwise you will likely be killed by a surprise attack. The exception to no dot on the map is where there is a major enemy that needs to be killed to complete a chapter (or a segment of that chapter).
Enemies tend to be more accurate, and also there is a chance you can get killed with one hit (but this doesn't happen all to often, as long as you remember to keep in cover during firefights).
Sections with time limits give you less time.
Your life regenerates more slowly.
I am not a particularly brilliant gamer, but I completed this game on hard, for the most part, without too much of an issue. The main problem I had was in chapter 14, which I will help you with below. I will leave a gap as there is...
....................SPOILERS FOR CHAPTER 14!
After you've killed the guy that Eddie told you to kill, and lost the police and feds (make sure to either ditch your car or change its plates, and change your clothes to make sure you have no wanted level at all), you need to take Joe to a car he's got waiting. Once you've dropped off Joe, go to the gun shop in the northwest corner of the map (the one run by the ex-army guy who gives you the huge gun you get for the distillery mission). Stock up on all ammo and, most importantly... GRENADES! You'll probably need about $1500, so if you don't have that, crush some cars at Bruski's scrapyard until you have the cash.
Scroll further down to see the next bit. Again - SPOILERS!
For the second part of this chapter, you need to go to the docks to get some more work from Derek. A cutscene happens, and some striking dockworkers reveal that Vito's father was killed by Derek's assistant Steve (presumably under orders from Derek). This enrages Vito, and the next objective is to kill Derek and Steve.
The first part of this isn't to difficult - just keep well covered and take out all the enemies. You should be able to kill Steve early on if you're quick and accurate with your rifle.
Once you've killed all the enemies outside, proceed to the warehouse. The door to Derek's office is locked, so you have to go in via the open sliding door on the ground floor. BEFORE YOU ENTER THE WAREHOUSE, make sure you've equipped an automatic weapon, and RUN IN head to the back of the warehouse, via the left hand side. Two guys will spawn at the back by a pile of boxes, one on each side. Take out these guys quickly, as they are accurate and have automatic weapons. Once these two have been killed, walk back towards where the top bit finishes (the top floor is a canopy, which covers probably one third of the warehouse).
Be wary of heading too far out from the cover, as there is a number of Derek's goons in here, and Derek will launch molotov cocktails from the top, which have a large impact radius and can kill you quickly.
Make sure you're walking on the side that is on your left as your heading out. As you approach the end of the canopy, turn around so you're facing the back of the warehouse. Hold LT to arm your rifle, aim upwards and walk backwards slowly. As you emerge from underneath the canopy, you should see one of Derek's goons waiting there. If you're quick, you can gun him down before he spots you. Once you've killed him, take cover and make sure you can see the stairs, as two goons will come down after you. Take them out.
Once those goons are dead, go to wherever you need to be to be directly under the large red & white circle indicating Derek's position. ARM GRENADES. Once again, walk backwards so you're facing the canopy, once you're clear, launch a grenade and RUN back under the canopy. With a bit of luck, your grenade will kill derek. If not, repeat, but be warned as two more goons will head for the stairs so be ready to take them out.
It may take more than one grenade to finally get Derek, but you'll know when Derek is dead as Vito says a little speech about not following in his father's footsteps. You need to run back out and take cover near the entrance as there may be one or two goons left upstairs, but as long as you keep in cover you should be able to take them out without too many problems.
Sorry for how wordy that is, but it's without a doubt the most difficult bit of the game. It's even tough to beat it on easy!