There are some vague references to Dark Souls One, and the main antagonist came to be as a byproduct of something that happens in the first game, but really, you wouldn't be missing out on anything starting with the second game. The way the story is in Dark Souls, things aren't really explained to you, you kinda have to talk to all the NPCs and read item descriptions and use the information you get to put the pieces together.

A warning though if you plan on getting it, it can also be confusing what you are supposed to do if you don't have someone to tell you what areas you are supposed to go to first and what some things do. And there are a few parts/areas/enemies that aren't just hard, they are unfair. Like I said, as fun of a game as I think it is (When you aren't ripping your hair out with frustration at how hard it can be), it has its flaws.