This will take you all the way down to the chapel just before the Dragonslayer Armor bonfire. Once you’ve got to the top of the steps make a hard right to another lift. If you’re doing the latter then stay roughly in the centre while going through the barricades, then hard right up the stairs to avoid attacks from the knights (although you’ll still need to do a lot of dodge rolling). You can either fight all of them or opt to run through. The entrance to the boss area is in the distant building opposite, which requires you to run through a bunch of hollows, up some stairs, and then through a bunch of knights.
From the top of the lift go right, which takes you outside onto the walkway. You need to have opened up the last shortcut at the Grand Archives, which opens up the lift from the bottom of the archives all the way to the top. The fog door for the Twin Princes is miles away from the nearest bonfire.
This can be one of the more frustrating fights of the game, but you’ve come this far, right? 1. So they seem well balanced, as you can imagine.
Spoilers, but the Twin Princes are the penultimate boss fight, and the result of a royal family inbreeding for years and then, when that failed to produce a good enough heir, performing terrible and unspecified experiments on their children.