You'll be rewarded with the achievement once you complete all the storyline relative quests of the Assassin Guild.
Contracts on Narfi, Ennodius Papius and Beitild*** Spoiler - click to reveal ***Nazir will confide these easy kills to you. See it as practice and try to figure out what you can do in highly frequented cities. There are very numerous ways to take out Beitild in Dawnstar and get away with it. Beside the classical 'be silent and discreet', one way would be to tell her you are from the Dark Brotherhood and make her pursue you out of the city and out of sight. Another would be to have the Silent Cast perk and use a frenzy like spell on her, so the guards come at her and do your job. Be creative! Return to Nazir once all three are dead, collect your gold reward so you can go on to the next quest.
Mourning Never Comes*** Spoiler - click to reveal ***Astrid will send you to talk to Muiri in Markarth, who will give you a contract on Alain Dufont. Once done, return to Muiri and then Astrid to move on. You will also have the possibility to deal with Nilsine Shatter-Shield, but know that is totally optional.
Whispers in the Dark*** Spoiler - click to reveal ***Astrid wants you to spy on Cicero. Enter the Night's Mother coffin and enjoy the scenes. Tell her what you have learned afterwards and she will send you to Nazir to get you busy a while.
Contracts on Lurburk and Hern
The Silence has been Broken*** Spoiler - click to reveal ***Astrid finally agrees to your suggestion, so you have to go to Voluunrund to talk to Amaund Motierre. Then return to Astrid, then go to the Ragged Flagon to find Delvin Mallory (whom you may already know) aaaand... come back to Astrid.
Bound Until Death*** Spoiler - click to reveal ***This one is a real mission: killing a bride at her wedding! You will even get a bonus if she is addressing the crowd when she dies. There are many ways to get this done, but fast traveling before anyone becomes hostile is a cheap getaway. Otherwise a fun way to accomplish it is to use the gargoyle statue over the balcony; speak to Babette about it if you're interested. Getting her while she is talking from the balcony will grand you the amazing gold bonus.