Here is the full list of all 41 DARK SOULS: REMASTERED achievements. It takes around 60-80 hours to unlock all of the achievements on Windows.
Discover Princess's Guard covenant.
Acquire the Lordvessel.
Arrive in Anor Londo.
Discover Chaos Servant covenant.
Ring Bell of Awakening in Quelaag's domain.
Ring Bell of Awakening at Undead Church.
Discover Way of White covenant.
Arrive in Lordran.
Acquire Estus Flask.
Light bonfire flame.
All achievements completed. Congratulations!
Reach "To Link the Fire" ending.
Reach "The Dark Lord" ending.
Acquire all rare weapons.
Acquire all sorceries.
Acquire all pyromancies.
Acquire all miracles.
Discover Blade of the Darkmoon covenant.
Discover Warrior of Sunlight covenant.
Discover Forest Hunter covenant.
Discover Darkwraith covenant.
Discover Path of the Dragon covenant.
Discover Gravelord Servant covenant.
Acquire best weapon through standard reinforcement.
Acquire best weapon through crystal reinforcement.
Acquire best weapon through lightning reinforcement.
Acquire best weapon through raw reinforcement.
Acquire best weapon through magic reinforcement.
Acquire best weapon through enchanted reinforcement.
Acquire best weapon through divine reinforcement.
Acquire best weapon through occult reinforcement.
Acquire best weapon through fire reinforcement.
Acquire best weapon through chaos reinforcement.
Acquire the Rite of Kindling.
Acquire the Art of Abysswalking.
Defeat the Soul Lord Gravelord Nito.
Defeat the Soul Lord Bed of Chaos.
Defeat the Four Kings, inheritors of souls.
Defeat Seath the Scaleless, inheritors of souls.
Defeat Dark Sun Gwyndolin, the Darkmoon God.
Defeat Crossbreed Priscilla, the Lifehunter