Complete two full playthroughs of the game, earning the two different endings.
Obtain all Rare weapons (requires a shortened third playthrough)
Obtain all 24 Sorceries
Obtain all 19 Pyromancies
Obtain all 23 Miracles.
Discover all seven covenants.
Defeat two optional bosses.
Fully upgrade weapons on the 10 main upgrade paths.
Everything else will come through natural progression. See individual achievement solutions for more detailed information.
Note that the remaster includes content that was originally added via DLC on the 360 version, but nothing from that content is required for achievement completion.
And if you're interested, this site
offers a workable checklist that basically tells you what to do as you progress through each area.
As always, refer to http://darksouls.wikidot.com
for specific information relating to items or locations.