The Death Arcana
Players unlock the Death Arcana relatively early on in the game since the person it is tied to is the Mysterious Boy. The same one that had you sign the contract when you first arrived at the Dorm. This is one of the few Social Links within the game that you're unable to advance on your own. Throughout the game the mysterious boy will show up to speak to you, usually waking you up while you sleep to share some information with you/talk with you. These are the times that the Social Link will raise.
You should expect the Death Social Link to cap itself out somewhere around the middle or 3/4 of the way through the game. Once you've forged the bond that cannot be broken with this Social Link you'll be able to create the Persona Thanatos.
While raising a Social Link you'll be given story cutscenes where you'll have to answer questions. The answers to each of these questions is different depending on the person and the situation. What you usually need to try and do is pick an answer that reflects their personality. Here's a cheat sheet for each Social Link rank of the Death Arcana and what to say.
Rank 1 - Rank 10 all automatically earned
When you complete the Death Social Link you'll be able to fuse the Persona Thanatos through Hexagon fusion. You'll need Alice, Mot, Samael, Loa, Pale Rider and Ghoul in order to perform this fusion.