Achievement progress is synced with Xbox live automatically. if you unlock an achievement it is not saved to your game save, rather it is saved to your systems local achievement DB and this will get synced with XBL. So deleting your game save will not erase this kind of achievement progress.
for achievements like complete the game on hard or kill 1000 enemies then deleting your game save before unlocking the achievement will mean that you will need to start again from scratch to get back the point you were at before. Some games will have network tracking of kills and some other count related achievements. If your game save is deleted this progress will remained synced on Xbox Live