To unlock Hardcore Mode you will have to complete the game on Hard difficulty. Then start a new campaign and both Tier 1 and Hardcore difficulties will be available.
In general Hardcore difficulty is on level with Hard difficulty. The only difference is that you'll have to restart the whole campaign if you die on Hardcore...yup, you read that right. Not just the mission...the whole bloody campaign...
...there is a very easy and effective workaround that'll make it a breeze anyway.
What you need is a USB Stick and a bit of patience.
The game doesn't let you start from the last checkpoint if you die. You still get checkpoints though. If you quit to menu and come back to the game later you can actually continue from the last checkpoint you got...and that is what you'll use to your advantage.
When you got a checkpoint in the game you will quit to menu and then quit the game. It is important to quit to menu first otherwise you will not get a save. Once you're on the dashboard you'll go to Saved Data Utility (PS3) and copy your savegame from HDD to your USB Stick. With that you will have your last checkpoint saved. Now go back to the game and continue the campaign like nothing happened. If you die now you simply quit the game and go to Saved Data Utility (PS3) again. Now copy that old savegame from your USB Stick to your HDD and you'll have your last progress status restored.
There are only two missions in which this method doesn't work and those are the ones you drive the cars in. They aren't really difficult and shouldn't give you any trouble.
How often you save and backup those checkpoints is completely up to you. You can save every single one you get or just do it at the beginning of every level. Just make sure not to mix up the two saves and you're golden.
Thanks to FedderTheGun for contributing to this solution.