My only successful no death run on a new character went something like this.

This is the easiest order IMO. They're all numbered in a sense that you will progress in levels and/or items enough to pass the areas and bosses with the least difficulty, while still being doable on your first playthrough.

1. The Last Giant - skip pursuer, unless you can find a summon (or you're already a freakin' god).
2. Dragonrider (proceed to Hunstman's Wharf bonfire, then travel back)
3. Old Dragonslayer (I like to kill Targray for the Blue Knight Halberd, it's high base damage, range and armour negation rips through mobs and bosses early if used well, and doesn't need to be upgraded.)
4. Flexile Sentry
5. Skeleton Lords - proceed as far as Harvest Valley bonfire (avoid fighting the Mounted Overseers, dying in Earthen Peak is too easy so be careful travelling forward)
6. The Pursuer
7. Ruin Sentinels
8. Executioner's Chariot, a.k.a. the only use for alluring skulls
9. Covetous Demon
10. Belfry Gargoyles
11. The Lost Sinner
12. Scorpioness Najka
13. Royal Rat Authority
14. Prowling Magus and the Congregation
15. Mytha, the Baneful Queen (proceed to bridge bonfire)
16. Duke's Dear Freja