Life on Mazam
When her world was conquered by the Daleks, she agreed to surrender all weapons in order to save her people. When this was done, the Daleks decided to exterminate her as she was no longer of use to them. Before they could do so, Abslom Daak materialised in the room and killed the two Daleks present. Together they escaped from the palace and attacked the Dalek base nearby. Taiyin and Daak fell in love with each other. The pair planned to run away to a peaceful planet using Taiyin's space yacht, but she was shot by a stray Dalek survivor and died in Daak's arms. (COMIC: Abslom Daak... Dalek Killer)
The pair knew each other for less than a day, but Daak would spend decades idolising her memory. (COMIC: Physician, Heal Thyself) Quickly moving to preserve her life, Daak kept Taiyin's body in a cryogenic freezer-unit, hoping to someday find a medic capable of reviving her. To Daak's recollection, it took about 20 minutes from her death to entering the unit. (COMIC: Star Tigers, The Then and the Now) This unit was kept in Taiyin's space yacht, which Daak used to escape Mazam, ultimately landing on Draconia. On Draconia, Taiyin was examined by Draconian doctor Zorynx, who determined that her potential revival was beyond the range of Draconian medical science. Leaving Draconia, Taiyin's unit was transfered to the Kill-Wagon, a prototype Imperial-class Frontier Defence Cruiser belonging to Prince Salander of Draconia. (COMIC: Star Tigers)
Ultimately, the Kill-Wagon was shot down and crash landed on the planet Hell, leaving Daak as the sole survivor of his crew. Taiyin's unit also survived, and was found by the Seventh Doctor, who deemed her to be dead. Daak himself seemingly perished as he destroyed the Daleks' Death Wheel. In gratitude, the native Helkans prepared a marked grave in which they buried Taiyin's body on the surface of Hell. (COMIC: Nemesis of the Daleks)
Later, Taiyin's body, still in the cyrogenic unit, was uncovered on Hell while Daak's crew were revived, supposedly by Helkan medicine. Just before his death, Daak was transmated to safety by the Emperor Dalek. Led to believe he had been saved by humans, Daak was given the task of finding the Doctor in return for Taiyin's revival. Capturing the Seventh Doctor and Bernice Summerfield on Hell, Daak's camp was transmated back to his benefactors, who had Taiyin's body taken to a special unit. Daak realised too late that he was actually on Skaro, and his benefactors were Dalek robots. Daak soon found Taiyin's body out of her cyrogenic unit, actually a trap prepared from the Daleks who had Taiyin's mouth emit gas which knocked out Daak and his company.
After being made to fight Davros' Daleks on Spiridon, Daak returned to Skaro, where Davros usurped the Emperor Dalek. Daak was intent on rescuing Taiyin, though Benny told him that she was dead, urging him to let her rest in peace. When Daak insisted on staying behind, the Doctor had his crew force him into the TARDIS. Following his ordeal, the Doctor believed that Daak had gotten over his fixation with Taiyin and had perhaps found another as he began to develop feelings for Benny. (COMIC: Emperor of the Daleks!)
Following the Last Great Time War, however, Daak was travelling with Taiyin's unit again. With no Daleks to kill, Daak took up a job to bring the Doctor to a judge for a trial. Using his vortex manipulator, Daak tracked down the Eleventh Doctor and occupied his TARDIS. In response to Daak's threat, the TARDIS hid Taiyin within its structure. Daak was then forced to join the Doctor in his quest to discover who killed the Cyclors. (COMIC: The Then and the Now)
After Daak was shot by the Squire (COMIC: Fast Asleep, Gently Pulls the Strings) Daak saw a vision of Taiyin, who hooked him up to a machine that healed his wounds. Daak then exited the TARDIS on a speeder and ran over the Squire. The Doctor then granted Daak his greatest wish, and used the Then and the Now to send him, ever accompanied by Taiyin's corpse, into the Last Great Time War, where he could slay Daleks to his heart's content. (COMIC: Physician, Heal Thyself)
Behind the scenes
While many early issues to feature Abslom Daak would treat his fixation on Taiyin as a romantic disposition that drives his hatred for Daleks, many stories written decades later would treat his feelings for Taiyin as an obsession. Several times, writers would attempt to illustrate Daak "moving on" and even burying the body of Taiyin.
In interviews with altered vistas, Steve Moore (the writer of many of the early appearances of Daak and Taiyin) stated that he had not been happy with how later issues of DWM had used each character.
...generally it seems to me that later editors and authors have overlooked the extreme importance of Taiyin to the story, especially now that she’s dead. Without her, Daak is just a thug with a chainsaw; with her, he’s a tragic figure who has real depth of character. Unfortunately it seems that most other writers and editors just want the thug with the chainsaw, which just doesn’t go anywhere and certainly doesn’t fit with the more idiosyncratic world of the regular Doctor Who stories, so it’s hardly surprising Daak’s become a figure of fun in later appearances.
Moore died before the publication of Doctor Who: The Eleventh Doctor Year 2, which more closely stuck to the "basics" of their relationship.