Originally i wrote this solution as I was getting the achievement, I now have the achievement and there are a few things i want to point out. I noticed some confusion on this topic in the solutions so I wanted to confirm that you have until the end of Eden Prime to change the difficulty, also something that confused me is that after you beat it on veteran or normal in the options it will say that the hardcore difficulty still must be unlocked. Select it anyway and save changes.
In order to cope with the difficulty I would suggest using the infinite money glitch (the med clinic one) to get the money needed to buy all HWP spectre gear. It is NOT required to do this I just figured i would point it out as it can help loads with the difficult enemies. Whether or not you decide to use this method is up to you, however I find that putting heat sinks into the weapons makes them even more effective. This should help a lot facing the enemies. My recommendation is to keep a biotic or at least a partial one in your party at all times. Frequently it is easier to lift an enemy or throw them into oblivion or a pit rather than unloading into them for 15+ seconds. (I'm looking at you Thorian creepers). Lift and throw combos work especially well against Krogan and Geth Destroyers that charge you constantly. In addition, polonium or chemical rounds negate the Krogan's health regen for the most part. Save often and do your best to try not to get mobbed by swarms of enemies. I also was going for the Ally achievements at the same time as this one and because they require you to complete a majority of the game with the same squad members I ended up doing most of the side quests. This will net you a lot of exp for leveling although it is not required. In addition this achievement can be gotten with any class. Most people prefer soldier but I got mine with an infiltrator so it is possible. With regards to the infinite money cheat, occasionally you will lose the ability to buy back your special ammo types (polonium, incendiary, and cryo rounds), this is not too big of an issue as you will get better ones quickly just by progressing through the story.