Looking at that list of multiplayer achievements and based on my own personal experience online, the majority of them will be very easy to get through regular play. Here's some tips on getting some of the trickier ones.
Mastering the Art:
Earn the INCOGNITO bonus (Multiplayer).
You need to get an incognito kill. Depending on your skill level, this could be a tough one to get. To get an incognito kill, you must fill up your approach meter to the max by keeping your target within your line of sight without running up to it or otherwise making yourself noticeable. Note that you can run without reducing the meter if they are not within your line of sight.
Make the Headlines/Tactician:
Obtain 12 different Accolades (Multiplayer).
Score at least 2500 points in a session (Multiplayer).
To be successful in multiplayer, you need to always go for the quality kills. You won't necessarily win by getting the most kills, you win by getting the most points and kills only grant you 100 points. If you are more patient, you can get bonuses to earn you even more points. Being fast an efficient will get you this achievement naturally.
This could be tough getting because you will need to be quick at getting your quality kills while still fending off pursuers. I can't really help much with this, you just gotta be a machine and some other players can be very difficult to beat. Though out of all the game types, Deathmatch might be the easiest to get it in as everyone is in a confined space and there aren't (normally) any lookalikes so should be easier to find your targets.
The other achievements you should be able to naturally earn if you go after the True Templar achievement of level 20. Depending on your skill, it might take a couple of days or so of playing. You shouldn't strain yourself over this too much if you plan on playing a lot.
Unfortunately, there are bugs with the co-op mode in LEGO Harry Potter: Years 1-4, which can result in any of the "buy all types of characters" trophies not rewarding properly. I hate to say this, but the only solution is to restart the game from a fresh save and ensure only one player purchases the characters.
From PS3Trophies.org:
Q: Help! My Wonderful Weasly/Quidditch Team/Dark Wizards/Animagus Trophy didnt unlock! Why? What can I do?
A: These trophies are known to glitch if any of the characters needed
for the trophy are purchased by the player 2 in co-op mode. There is
no known remedy for the situation so far aside from starting the game
over for that trophy. The best way to ensure these trophies unlock
normally is to make sure player 1 purchases them all. Sometimes it is
even better to be safe and have player 2 (if playing Co-op) drop out
while you purchase characters.
Best Answer
A quick duckduckgo search provided :Link To Guide With Items
Chaos Blade
Crystal Ring Shield
Darkmoon Bow
Dragonbone Fist
Dragonslayer Spear
Golem Axe
Great lord Greatsword
Greatshield of Artorias
Greatsword of Artorias
(True) Greatsword of Artorias
Lifehunt Scythe
Moonlight Butterfly Horn
Quelaag's Furysword
Smough's Hammer
Tin Darkmoon Catalyst
--Weapons or Shields dropped by enemies--
Black Knight Greataxe
Black knight Greatsword
Black Knight Halberd
Black Knight Shield
Black knight Sword
Channeler's Trident
Crest Shield
Crescent Axe
Demon's Spear
Giant's Halberd
Grant
Sanctus
Silver Knight Shield
Silver Knight Spear
Silver Knight Straight Sword
Stone Greatshield
Stone Greatsword
--Weapons found as treasure--
Astoria's Straight Sword
Blood Shield
Dragon Crest Shield
Dragon Tooth
Dragonslayer Greatbow
Effigy Shield
Havel's Greatshield
Velka's Rapier
--Weapons or Shields acquired from covenants--
Dark Hand
Gravelord Sword
Dragon Greatsword
-- Weapons acquired from cutting off boss tails--
Drake Sword
Dragonking Greataxe
Moonlight Greatsword
Priscilla's Dagger