To speak to Lucien Lachance and get the chance to join the Dark Brotherhood, you need to murder any innocent civilian in Oblivion Remastered. Once you've murdered this civilian, take a rest in a bed that's located in a safe spot with no other NPCs around. During your slumber, Lucien Lachance will wake you up and speak to you about joining the Dark Brotherhood.
In terms of the civilian you want to murder, it's best to avoid any quest characters, as you might need them later on in Oblivion Remastered. The optimal choice is a random NPC who isn't inside a large city, as that murder would place a bounty on you. Instead, you want an NPC who isn't near any guard and ideally is alone, so you don't have to murder anyone else that's nearby.
After you pick your prey, murder them in any way you see fit, and then find your bed. You'll be woken up automatically by the game to speak to Lucien Lachance, who fills you in on the task you need to complete to join the Dark Brotherhood.
Completing Lucien Lachance's quest in Oblivion Remastered
Once you're done speaking with Lucien Lachance, you'll learn that he wants you to assassinate a civilian by the name of Rufio. Lachance gives you the "Blade of Woe" sword to do the job. This begins the quest called "Knife in the Dark" in your journal. Rufio is located at the Inn of Ill Omen, which is an inn found northwest of Bravil and east of Skingrad along the Green Road.