Below is simply a brief (though my no means comprehensive) run-down of some of the more important relationships Bellatrix Black Lestrange was involved in, in both Canon and Fanon.

Bellatrix & Andromeda
Who?: Andromeda Tonks, nee Black.
Canon or Fanon?: Both
Canon Relationship: Sisters.

Unlike Bellatrix, and their other sister, Narcissa, Andromeda did not believe as the rest of her family did, that Muggles and "Mud Bloods" were unworthy to even socialise with. It is assumed that Andromeda and Bellatrix may have gotten along as very small children, before they developed different ideas about the world. When Andromeda began her schooling at Hogwarts, she was sorted into Hufflepuff House, which may have strained the relationship between the two sisters even more, as many believe (though it is not directly stated) that Bellatrix was a member of Slytherin House. When Andromeda married a Muggle, then had with that Muggle (Ted Tonks), a daughter, Bellatrix, who perhaps had already began to train for her role as a Death Eater, most likely drew even further away from her sister. Indeed, even in Fanfiction, interaction between Andromeda and Bellatrix is quite rare, and, when it does occur, is rather hostile.

Bellatrix & Lucius
Who?: Lucius Malfoy
Canon or Fanon?: Both, mostly Fanon.
Canon Relationship: Siblings-in-Law.

Lucius Malfoy and Bellatrix are perhaps more alike than any other two characters in the Harry Potter series. Lucius and Bellatrix were both born into powerful, pure-blood Wizarding families, both held many of the same beliefs, and both, after having graduated Hogwarts, became Death Eaters. However, one point on which Lucius and Bellatrix differ is Loyalty; Bellatrix is extremly loyal to Voldemort, going to far as to welcome a stay in Azkaban Prison while defending her crimes, whereas Lucius avoided such a sentence by saying he had worked under force of the Imperius Curse.

While Lucius married Bellatrix's sister, Narcissa, many scenarios in Fanfiction depict Lucius and Bellatrix has having had some sort of relationship during their Hogwarts years, with Narcissa perhaps none-the-wiser.

Bellatrix & Narcissa
Who?: Narcissa Malfoy, nee Black.
Canon or Fanon?: Both
Canon Relationship: Sisters.

Narcissa Black Malfoy is sister to Bellatrix; Unlike their other sister, Narcissa believed, as Bellatrix did, in the ideals of Voldemort, and believed themselves, as Slytherins, Blacks and Pure-Bloods, to be better than most anyone else. Therefore, it can be generally assumed that Narcissa and Bellatrix got along fairly well as children and young adults, if not the best of friends.

In Canon, no evidence is directly given as to how often Bellatrix and Narcissa communicate, though Fanon oft depicts the sisters as very close. An incestuous relationship between the two sisters is a fairly popular theme in Fanfiction.

Bellatrix & Rodolphus
Who?: Rodolphus Lestrange
Canon or Fanon?: Both
Canon Relationship: Husband and Wife.

While it is unclear exactly when Rodolphus and Bellatrix were wed, many assume the event occured not long after both had graduated from Hogwarts; Like Bellatrix, it is assumed that Rodolphus also was a member of Slytherin House during his time at Hogwarts; Also, like Bellatrix, Rodolphus became a member of Voldemort's Death Eaters, shortly after graduation. Perhaps this, their involvement in the Death Eaters, can be seen as the reason why the Lestranges never had children. Rodolphus, and his brother Rabastan, accompanied Bellatrix in the search for their missing Lord, slinging Unforgivable Curses along the way. A popular theory in Fanfiction, interestingly, holds that the marriage between Rodolphus and Bellatrix was/is one of convenience, rather than one of love. This theory seems to stack up in DH, and JKR's interviews since then, when it is indicated that Bellatrix is loyal to her husband but loves Voldemort.

Bellatrix & Sirius
Who?: Sirius Black
Canon or Fanon?: Both
Canon Relationship: Cousins

Bellatrix was involved in perhaps no other relationship more volatile than the one she had with her cousin Sirius; Like Bellatrix's sister, Andromeda, Sirius did not fit the Black mold -- He, like Andromeda, did not believe in the things the Blacks had built their family on: Contempt for "Mud-bloods" and Muggles, violence, and intense, blinding arrogance. Bellatrix's outright hatred for Sirius developed, most likely, even more the two became adults, though it is assumed said hatred did not take on such violent properties until adulthood. The culmination of this hatred, of course, came during a battle of theDepartment of Mysteries, when Bellatrix killed her cousin Sirius.

- Adia