The game's campaign follows up the story of Black Ops and is set in the late 1980s and 2025. In the 1980s, the player switches control between Alex Mason and Frank Woods, two of the protagonists from Black Ops, while in 2025, the player assumes control of Mason's son, David (codenamed "Section").
I've gone through all the campaigns and they are all pretty self-contained. You don't have to worry about previous storylines to get through this one. You don't need to play the earlier ones.