If the player gets to second base on his hit, it is a double.If a player is on first base and goes to second base on a pitch during another player's at bat, it is called stealing second base.The lead runner is the runner at the base closest to home plate when there is more than one runner on base.
They are responsible for covering the are up to second base.
They are also responsible for covering all the area to their left that the first baseman cannot get to.
That would depend on how the runner was retired at second base.
For example, if the runner slipped and fell on his way to second base and the outfielder had the time to throw him out on a force play, the batter would not get credited with a base hit.
If they fail to touch second base (and run directly to first base without re-tagging second base) they can be called out on appeal.