The difficulty achievements DO stack, so you should begin playing the game from the hard difficulty right away. For most of the game, the only real difficulty comes from trying to figure out what to do to progress through the level, which is a factor that is unaffected by the selected difficulty setting. The only times where you'll notice the hard difficulty kick in is really in chapters 5 and 6, where you are forced to confront a lot of enemies in combat.
There are video walkthroughs posted to each of the corresponding chapter achievement guides, including one for ch.6 posted on the 'Survivor (Hard)' achievement guide. Check all of those out to help you complete the game on the hard difficulty.
For anyone that is either unable to see the videos, or would prefer a written walkthrough, Paranoia7 at GameFAQs has a detailed chapter by chapter walkthrough. Refer to the link below:
http://www.gamefaqs.com/xbox360/640660-amy/faqs/63664