Firstly, to unlock hardcore mode, you must first play through and complete all levels of Director's Cut, which will also give you a good understanding of the levels/bosses once again.
Secondly, the change in "Hardcore" mode, is the only way you can kill zombies, is which headshots, which isn't very difficult with a fully upgraded SMG, (Which is what I used, and made the game in hardcore VERY easy) they can also be killed with grenades, fortunately. As you know by playing Director's Cut previously (or if you haven't yet) the difference between Director's Cut and normal story mode is in Director's Cut, you only have 3 continues, where-as in Story, you basically have somewhat unlimited. Director's Cut also have extra little areas in levels to make them much longer/more difficult. Just like the other level achievements, it will not pop until the credits are done.
The only level I had much difficulty on was the final one, in which I had used up all my continues, so I had 0 left and was on my final life, but I would HIGHLY recommend a fully upgraded SMG for this mode, due to how great it is.
I'm not one for writing large guides on how to kill bosses, but as I played through the game on hardcore not so long ago, I'll try my best to remember their tactics and weaknesses;
Papa's Palace of Pain: Jasper - This boss fight is pretty straight forward, Jaspers main weakness is his overgrown brain, so just spam it as much as you can when you can, the only sort of attacks he'll do is have objects levitate around him, before throwing one by one at you, fortunately you can shoot these objects down as they're around him, which I did at the same time of shooting him, afterwards, he'll chase you around throwing one object time to time at you. His 2 other useless moves is 1, levitate you, which all you do is hit a bullseye to get down, making him vulnerable. The other is create giant object which you spam your shots at to make explode, making him once again vulnerable.
Naked Terror: Coco and Sindy - I found this boss even easier than the previous chapters boss, you cannot hurt Sindy (the big one) the only one you can harm is Coco (the small one), coco will at first jump down infront of sindy, which you have a split moment to attack and take down her health, which then she'll proceed to jump between objects, hiding behind them, which if you're using an SMG or something quick like that, you'll have her health down before she even gets close to you. The second sort, is she'll begin swinging in circles on the poles, launching herself between pole to pole as she makes her way to you, which you'll just have to simply shoot her as she's on the poles, which at this point Sindy will throw (breakable) barrels at you.
Ballistic Trauma: Screamer - This boss has a relatively short/easy pattern to remember, at first, she'll appear on the other side of the room (infront of you) in which she'll teleport to other parts of the room closer to you, slowly making her way, just simply shoot at her (her whole body, doesn't matter where you shoot on this boss) before she makes it to you and hurts you. The second part is she'll summon a line of zombies, and once again do the teleporting, but behind them, so simply and quickly see if you're able to spot her behind the zombies and shoot at her, killing all the zombies will be a waste of time on this mode. The third attack is she'll rush towards you in a group of four of her, but only one of them is the real one, you can spot who because the three fakes are see-through, where-as she isn't, she's regular like a person, shoot her before she's able to get you, rinse and repeat.
Carny: Nigel and Sebastian - This is quite an easy boss also, with an easy pattern to remember. First of all, he'll begin walking towards you, preparing an attack which leaves the guy attached to the chest vulnerable, so shoot at him as soon as you get an opening, to avoid being hurt, and to hurt him, which your character will run back, and you'll be chased and repeat the same steps. Also on a side note, keep shooting the little guy even if there's no attack segment, it won't so /as/ much, but it will still damage him quite a bit. After this, he will walk back over to a big pile of bodies, which you'll have a brief moment to once again shoot at the small guy before they throw something at you, and before the evil clowns reach you. Once the clowns are dealt with, there'll be a little bullseye mini-game, which after you just repeat all the above steps until he's dead.
Creeping Flesh: Meat Katie - In this boss fight, the main point of which you have to attack is her utter, just like the previous boss, even when she's not doing main attacking segments, just shoot the utter when you have the chance for that extra damage. All this boss will do is come up close to you 2-3 times like Nigel and Sebastian, and raise her weapon in which you spam-gun her utter, after 2-3 times of doing this, you'll run to the other side of the room, and she'll charge at you, activating a bullseye mini game, easily rinse and repeat and you're done in no time.
Scream Train: Crawler - This boss can be quite a pain in the backside due to the angles you have to try and finish an attacking segment at before he attacks you, but take note his main weakness is his bulb-eyes, which aren't hard to miss, at all. First few moves, he'll be attached to the side of the train, raising his leg-like-blades, in which case you have to shoot quickly and accurately at his head, as it's a pain to get it done right at this angle, after a few times of doing this, like Meat Katie, he will charge at you, activating a bullseye minigame. Eventually after repeating these steps and weakening him enough, he'll hop on the front of the train with you, guarding his face before he prepares an attack, which you have to shoot his face to stun him, then shoot at his eyes to destroy one, and repeat for the other.
The Fetid Waters: Lobber - This is a very short and simple boss battle, basically he will hold big ball-like objects in his hands, preparing them to throw at you, just shoot them as he holds them to cause them to explode, damaging him, when done enough times, your character will move area, as you're being chased by bats of which you kill, sometimes the lobber will change it up by holding 2 at once, or just holding one giant ball, in which case, you just continue with your shooting strategy, after 2-3 are moves, he should easily be done. I had absolutely no trouble with this boss.
Jailhouse Judgement: Brutus the Ruthless - (Take note of the 2 health packages at the back of the room opposite side of each other on the silo like objects, and grenades down the middle section on the floor) I also found this boss to be very easy, take note of his weakspot that you mainly have to target, and can damage by shooting whenever, which is his right arm, the one that's basically a weapon and not normal, like the other. Basically, the first few times, he'll prepare an attack in which case you just shoot at his arm, sometimes he'll fire a bolt at you from his arm, which you can easily destroy. His second tactic will either be him collecting electricity from the wall as you stand opposite side of the room, he'll summon 3-4 plain zombies which you can easily headshot, then shoot at Brutus from across the room, his third tactic will be a bullseye minigame, which eventually leaves him pinned down with his arm stuck in the ground, giving you some time to just spam him with your bullets, rinse and repeat.
Overkill: Mother - This is probably one of the most difficult boss fights I had in the game, due to how long it was. The first part of which you fight her, you'll be in a tunnel like thing and she'll be chasing you, what I did what shoot at her chest/head area up until my minigun was just about to reach the red state, then let it cool, which by the time mother had stopped at the two pillars, I was able to quickly shoot at her again and finish off her "Boss Attack", which once done, shoot and destroy either (or both) the pillars to weaken her, then just repeat the steps until done.
The second part of the mother battle, you'll be above ground in a parking-lot type area. she'll have a few bunch of moves that she'll do here, I can't remember the specific order but I'll list them all, when you're beside the helicopter, she'll pick up a box of barrels, and launch them at you, so simply shoot the three red barrels before they hit you, as you come out the other side of the helicopter, mother will bend other and... poop out 3 mutants (the ones that explode acid when they get too close/die), so just kill those mutants asap to save your health. the next tactic she'll do is the common attack one which you can do to damage her, is raise her leg as you start to look up at her, just shoot her chest/head as quickly as you can to not risk your health (these little segments I found to be very difficult), the next one, is you'll have a close up of her beautiful chest, and you'll have to quickly shoot it (quite easily too) to damage her. Then the next one, is there'll be a bullseye minigame, which once done, you can just shoot at her normally as she gets back up on her feet. One of the final tactics of this little part is she'll pick up huge rubble and launch it at you, so you just destroy it before it reaches you. (It'll always get close, but not close enough if you shoot at it accurately). Rinse and repeat until done.
Now simple wait until the credits are done for the trophy.