You must beat the game once to unlock this mode. I am not sure if you can do it after casual, but definitely normal or hard. Once you beat the game once, you must start a new game on hardcore.
It is a whole new game, not new game plus and you must prioritize experience as you will not be able to unlock all power ups.
I recommend upgrading the weapons first then magic. The extra health is really helpful. Before you manually save hit start -> game and options -> game and double check it is still on hardcore. My game randomly changed it to casual twice. Once after failing a challenge and once after dying to the hydra.
There are really no more hints as the game is pretty linear. I recommend collecting the secrets again as you need all the help and extra experience you can get. Just keep at it and abuse the manual save system. It can frustrating, but power through and the achievement is yours.