You must first complete the game on Normal mode to unlock Hard. Then complete Hard to unlock Non-Stop ∞ Climax. Your awards for each chapter do not matter, feel free to die a lot, abuse lollipops, and get a Stone award for every level if you just want the achievement.
On Hard difficulty enemies suffer less hitstun, attack quicker, do more damage, and enemies encountered are heavily altered.
Non-Stop ∞ Climax is mostly identical to Hard, but almost all sources of Witch Time are completely disabled. Selene's Light and Bracelet of Time still work, but their magic cost has been considerably increased and they're situationally useful at best. They're no quick and easy tips to help you, you just have to git gud and practice.
The most difficult spots to merely complete even given abundant continues are the fight against Jeanne on the Sky Fortress, any verse with 3 or so so Joy's or Gracious/Glorious, and to a lesser extent even the final boss. Many Alfheims are far worse, but you can skip them.