you need permit for both the door and the deck
you can go to homedepot, they can design the deck for you for free and print you out a BOM , at that point you know pretty much the material costs
i would recommend that you build the deck yourself, it is really fun and the most basic framing "project" around a house. however, if you have the cash and/or have no interest whatsoever, you can get handy man to build it. labor costs you 1 to 2 times the material.
like i said, you need a permit, otherwise if city knows it, you screwed up big time.