Adding a pergola to my deck; ok to replace corner railings posts with pergola posts?
I would build it around, not connected to, your deck.  

Bury your posts 4 feet in the ground  in-cased in concrete, Use a post protectors around them for longer life. ( With the post protectors you do not have to use concrete, i would. )

This will insure that racking is minimal with the lack of room for knee bracing that  you have and what little racking there may be would not be transferred to your deck, which could weaken/loosen your deck over time. 

Added bonus: You could make it bigger, wider, taller; extend it out from the house/deck further. Think Larger deck in the future, more grapes.