Let's say my Steam Library is shared with person B. If I own Skyrim, and he plays it, and then I buy Dawnguard, can he play Dawnguard as well? Or will he have to buy his own copy?
More information about Steam Family Sharing can be found here.
On the DLC faq, on that page, it states:
Who owns and can access the DLC and in-game content associated with a shared title?
A guest will have access to the lender's DLC, but only if the guest doesn't also own the base game. Guests may not purchase DLC for a base game they don't own. Any player may purchase, trade, earn, or otherwise acquire in-game content while playing a game, but in-game items cannot be shared between accounts. These items remain the property of the account that purchased or acquired them, whether borrowing or lending the base game.
So the guest can play your DLC, but can't buy DLC unless they already own Skyrim.
It's pretty straight forward.