You start out on a mission to dure the curse, and then you get unwillingly roped into a quest to find Vendrick. But you are told that Vendrick understands the nature of the Soul, so it makes sense that your journey to find him would be tied to your search for a cure. As you go you learn that Vendrick was supposed to link the Fire using the Throne of Want but he decided against it. And now you have Nashandra trying to persuade you to replace Vendrick and link the Fire yourself. You keep going and you start learning that Nashandra may not be such a great person, and maybe you shouldn't link the Fire. Nashandra wants to usurp the Flame.

Vendrick, in his memory, sends you out to find the crowns of past rulers. The reason behind this is that those crowns hold the residual strength of those rulers, and if you claim them you could harness their strength. So you bring him the crowns, including his own, and he unlocks the strength hidden within them. This strength allows you to keep a hold of your Humanity and stop yourself from going Hollow, but it also proves you are worthy to link the Fire. So that's how the linking the Fire plotline ties in with the curing the curse plotline.

The two central themes of DkS2's story are "strength" and "monarchy", and they kind of go hand in hand. Vendrick had the strength to build a kingdom, the strength to lead his people and the strength to fight the Giants, but he lacked the strength to link the Fire. And since DkS2 associates being a monarch with linking the Fire, this means he wasn't truly a king. Incidentally, neither were the 3 DLC kings. Each had considerable strength, which is still contained in their crowns, but all of them eventually died and never linked the Fire, so none of them were true monarchs either. The player character's journey in DkS2 begins with them as a weak Undead who should by all accounts go Hollow very soon, to an Undead replete with strength and the capacity to link the Fire.

Another thing to consider is that we're basically just a pawn in a huge struggle for power. We have the Emerald Herald who wants us to link the Fire. We have Nashandra who seems to also want that, but secretly she wants us to usurp the Flame. We have Aldia who wants us to leave the First Flame and find an alternative. And we have Vendrick who previously wanted to link the Fire, but then became a pawn in Nashandra's plan, and now seems to want us to link the Fire even though he has his own doubts. We're an Undead with considerable strength, so all these parties want us to fulfill their missions. The curse curing mission brought us to Drangleic, and then all the others seized the opportunity to try and win us to their side.