This is just an additional bit of info to go with the 2 good guides already posted.
If,like me,you feel you have killed all the spider droids & can't see any more-try this.I went looking into the next,bigger room thinking there maybe some in there.There wasn't & I hit a checkpoint so I thought that I was going to have to replay the entire level.I exited to main menu & continued my game just to check & it did start me off in the bigger room but when I back-tracked to the '3 tank room' all the Spider droids had respawned making it easier to systematically kill them all & get the achievement.