Neon Genesis Evangelion devotes the plot of an episode to Kaworu Nagisa, the only person to ever explicitly show Shinji Ikari love, who only appears in this very episode. Yet this person's relevance to the plot is that he is the last Angel, meaning that Shinji has to kill his new best friend.
In Black Butler, Angelina Durless appears to be an important major character, only to be murdered by her own butler (who also happened to be a Shinigami), complete with a half-chapter montage of her story.
Asuma Sarutobi in Naruto has little to do until after the time skip. Even then, he doesn't get much "screen" time in the manga until a chapter or so before he's killed. It was lengthy and dramatic too. The anime's giving him a role in the Temple of Fire filler arc seems to have been an attempt to defuse this trope.
Likewise, Jiraiya appears in the Three-Tails Filler arc and gets a fair amount of screentime early in arc despite not being present very often for much of Part II. In the manga, he reappears in the Hunt for Uchiha Arc to infiltrate the Hidden Rain Village and find Pain, and dies in the course of doing so.
Danzo, whose only significant onscreen involvement in the plot until then was to get Sai enrolled into Kakashi's team, takes centre stage in the midst of the Pain invasion arc, plotting to overthrow Tsunade by not intervening to protect the village, kills the messenger frog to prevent Naruto from returning, gets himself appointed as the Sixth Hokage with a Breaking Speech, promptly orders Sasuke's execution, attends the Five Kage meeting, reveals what was underneath his bandages, attempts to take control of the ninja alliance, and is promptly disposed off by Sasuke with some help from Madara. All in the space of about 20 chapters or so (if you would exclude the lengthy flashbacks and discussions that Naruto and Nagato have).
A recurring theme as far as villainous characters are concerned:
Sasori, Deidara, Nagato and Konan all got flashbacks shortly before their deaths (in Nagato's case, he actually narrated his), as did the aforementioned Danzo.
Kisame did not, but that turned out to be a hint that he was Faking the Dead and not long afterwards he played it straight.
The enslaved zombies of the Edo Tensei undead army are already dead, but they sometimes get the spotlight shortly before their defeat.
Orochimaru and Itachi have averted this, as they get or appear in numerous flashbacks before and after their respective "deaths", but oddly not during (Oro got a very brief one in his fight with Sasuke, though he wasn't technically dead and still isn't); in both cases their character arcs and influence on the story continued afterwards, and they were major if rarely seen recurring characters.
To the annoyance of many fans, averted with Hidan and Kakuzu, and less controversially the Sound Four.
Battle Royale, in which most characters die within a chapter or two of the point where the story starts to focus on them. Practically guaranteed if it starts talking about their past.
The Trigun episode "Paradise" focuses on Nicholas D. Wolfwood and ends with his death.
In the manga he gets two whole volumes to himself, with Vash only appearing right at the end. Needless to say, following the full reveal of his tragic backstory and the resolution of said backstory, he winds up dying after overdosing on the same chemicals that aged him and gave him his enhanced skills.
Legato also gets one of these in the manga, focusing on his traumatic past as a sex slave in a town of criminals and cowards, where he was rescued by Millions Knives and pledged himself to his service. Shortly after this spotlight story, Legato winds up dead.
Although L is a major character in Death Note, his death episode is the first glimpse the viewer gets of his childhood. The episode also focuses more on his thoughts, feelings and doubts than ever, whereas before he was single-mindedly devoted to exposing Kira. You just know something bad is about to happen!
Used very frequently in Mobile Suit Gundam. If a minor character is given his/her own episode, chances are they'll die very, very soon.
In s-CRY-ed, Kimishima and Scheris die in the episodes bearing their names.
The entirety of Bokurano. You can tell a pilot's the next one to go once he/she gets focused on. Although it was subverted for a little bit. The next chapter after Kana's death is Jun's, so everybody figures he'll die next. Then Yoko makes the contract, something she wasn't supposed to be able to do, and the next few chapters after that are her Day in the Limelight - then she gets shot in the head. After that the focus goes straight back to Jun.
Nuriko of Fushigi Yuugi fame. The infamous Episode 33 (or manga volume 8). And Episode 34, which was mostly spent eulogizing Nuriko.
Sailor Moon R has Sapphire. He never mounts any attacks against the Sailor Scouts at all, unlike Rubeus and Emerald before him. He is only seen talking to Emerald and Diamond. In the episode "Brotherly Love", he discovers Wiseman's plans, is injured by Wiseman, and revealed to have been in love with Prizma, and essentially does a Heel-Face Turn, before he attempts to warn his brother about Wiseman's treachery and is killed by Wiseman.
Episodes Twelve and Thirteen of Code Geass R2, with Episode Twelve acting as a breather; and contrast, to Shirley's death.
Fafner In The Azure does this several times, in conjunction with Dying Moment of Awesome. Shouko's and Mamoru's Heroic Sacrifice, Kouyou's and Sakura's assimilation, and Michio's suicide attack. Subverted with both Kouyou and Sakura in that both recover. Kouyou ends up becoming a Master-Type Festum anyway to protect the island, then he comes back in Meir form for the movie, piloting the Fafner Mark Vier, the Fafner he originally used. Finally, his Fafner is immediately trashed and his Meir core promptly vanishes.
Also subverted with Reim Lunettes and Lily Baskerville, who, after getting a chapter's worth of attention with each other, put himself in a death-like state after being non-fatally wounded and got shot in the head then recovers respectively.
When Team Dai-Gurren is trapped in the space ocean during episode 25 of Tengen Toppa Gurren Lagann, Kittan gets his time in the limelight. He is featured more than in other episodes and over the course of the episode we find that Kittan has insecurity about not being as good of a leader as Kamina and that he loves Yoko. Yoko also brings our attention to the fact that while "Simon pulled from the top; Kittan pushed from below" reminding us that Kittan was indeed a big part of the rebellion. All of this right before he leaves the ship to perform an epic Heroic Sacrifice in order to help Team Dai-Gurren escape the ocean.
Bleach villains tend to get an extensive flashback explaining their tragic backstory and taking up one or more issues/episodes. Usually this happens right in the middle of the climactic battle in which they are killed, and there's an explicit rule to justify it: when zanpaku-to of similar strengths clash, their wielders gain insight into their opponent's nature. Sometimes it's just a general sense of what the person is fighting for, other times it's an entire documentary of their life story.