Is there any way I can get these achievements after I beat the game? Perhaps a challenge/replay mode? Do I have to delete all of my progress from my account and replay it if I wish to acquire ALL of the achievements on Assassin's Creed 1?
Or should I play it again on my account without deleting it?
It's been a while and I don't remember how many files you can make for this game.
For reference, here are some solutions to the achievements over on TrueAchievements.com: Conversationalise; Hungerer of Knowledge. As Crimson Drifter suggests on his solution, Hungerer of Knowledge can be obtained after you beat the game by replaying memory blocks. Unfortunately, Conversationalise can only be gotten during a single playthrough and once you miss the occasion, it is gone.
What I would recommend is loading up your save and get the remaining memory glitches done. If you paid any attention to them at all, you should already have made good progress during your playthrough. After you have gotten that achievement out of the way, start a speed run of the game. You don't need to do anything special, just do the bare minimum before you can start the assassination missions and you should be fine. An exhaustive list of opportunities to talk to Lucy is also listed in iiM SO N1NJA's solution.
You can have 3 different saves simultaneously, so you don't need to delete your progress to play the game from the start. You may be able to get the achievement "Hungerer for Knowledge" by replaying memories via the DNA menu, but you will need to start a new game to unlock Conversationalist.
