What is the background story of Hatsune Miku? I do understand it is a fictional character, and so I feel there must be a story behind this virtual Diva. I am interested in learning "how" Hatsune Miku became interested in singing.
As is the case with most vocaloids, the company that invented 初音ミク (Hatsune Miku), Crypton Future Media, released an official personal data sheet for Miku; however, this contains her physical and technical traits rather than personality or back story, which intentionally allows the fans to take a role in developing the characterization of the character.  She is referred to as a "beloved collaboratively constructed cyber celebrity with a growing user community across the world." Some fans have been allowed to create officially-sponsored costumes for Miku through a design competition for the Sapporo Yuki Matsuri (Snow Festival).  It could be said that the clothing design becomes a part of the character's personality (i.e. this is the sort of fashion that Miku would wear).

『メーカー非公式 初音みっくす』 (Meekaa Hikoushiki Hatsune Mikkusu) is a manga about Hatsune Miku which ran in『月刊コミックラッシュ』(Gekkan Comic Rush) magaizine published by ジャイブ株式会社 (JIVE Ltd.). It is licensed in North America by Dark Horse Comics under the title Hatsune Miku: Unofficial Hatsune Mix.

『週刊はじめての初音ミク』 (Shuukan Hajimete no Hatsune Miku) is a yonkoma gag manga that ran in 『週刊ヤングジャンプ』(Shuukan Young Jump) magazine published by 集英社 (Shuueisha).

Additionally, some mangaka of 『りぼん』(Ribon) shoujo manga magazine published by Shuueisha have drawn illustrations of Miku that were included in a 付録 (furoku = freebie) included with an issue of the magazine, and on some special pages printed in the magazine. 

Each of these portrayals have different aspects and storylines for Miku, in keeping with the collaborative, undefined nature of vacaloid characters. There is no one background story for Miku. Crypton intentionally encourages fans to create their own songs, music videos, visual novels, doujinshi, fanfiction, fanart, etc. and basically anything is fair game; portrayals that strike a chord with many fans rise in popularity, where fans adopt another fan creator's rendition of Miku into their own image of her character. Miku is who each person wants her to be, which includes limitless options for coming up with her backstory.