Death is the incarnation of Death from the Discworld. Earth's Death was overworked, so the Interdimensional Guild of Death asked several Death's from around the dimensions to help. This Death was assigned Japan. Traveling between the dimensions was becoming something of a headache, so Death decided to take an apartment in Nerima.
Death lives, or rather resides, in apartment 4B. The extra three bedrooms are in case War, Pestilence, and Famine from the Discworld decide to visit. The apartment is decorated in black and grey, with a few touches of red for color. In general, the apartment looks like Salvador Dali decorated. Since we're talking about Death here, it's entirely possible Dali did decorate. The whole appearance of the apartment is somewhat macabre, accentuated by the eerie lighting. Death maintains the appearance by keeping his state of the art entertainment center in his "bedroom" (Death doesn't really sleep per se).
Death has reserved stable space for his horse Binky.