This is primarily opinion based, and this answer is based on my play style; which may or may not fit your play style or anyone else.
When I start a thief in Skyrim (which is the primary class that I play) I always play with a Bow. I find it fun to snipe enemies from afar and when you get the cool critical animation from hundreds of yards away, it is something to be satisfied with.
As far as skills and what I focus on first:
This is going to be one of your key skills in order to pull off effective backstabs, and other thieving abilities.
If you choose to go with the bow, level up the bow. If you choose to go with any other weapon, make sure you stick with that. Personally, I've tried the two weapon fighting style, and while it's fun, it can be a challenge when facing higher level creatures/people in the game. This is where Enchanting comes in.
This skill is a fun skill to level up, and it's quite easy; as long as you spend the time to do it. I found that Alchemy is a bit easier to level for me as there are more plant and edible things to eat around the world than Soul Gems
I don't really care for illusion spells, and as playing a pure thief, I find myself using less spells than using my bow.
I use the gear from the Thieve's guild as those give me the bonuses I want towards my thieving skills. Whereas The Dark Brotherhood will give me skills in tune with a magic based thief.
The Shadow Warrior Perk is legit. Here is a forum post that goes into greater detail on this skill. Get it, use it, survive.