If you are looking for Shostakovich online it is useful to keep in mind that his name can be rendered in Latin characters in a number of ways:
Schostakowisch, Dmitri Dimitriewisch
Šostakovič, D. D.
General biographical overviews can be found in virtually any encyclopedic source from the imdb website and wikipedia online to the Britannica, regardless of language. Certainly, subject specific sources like the Grove Dictionary of Music and Musicians have more extensive discussions and bibliography. The numerous bibliographic sources on Shostakovich all include references to biographical sources as well. If you are looking for Russian language material Справочная литература по музыке: Словари. Сборники биографии, и.т.д. provides the most complete list up to 1964. It can be supplemented with the volume of the same title that cover 1979-1995.
Shostakovich is one of those figures for whom you can find an entry in almost any reference source. So some general guidlines. If you are just looking for an overview on his life, either in English or Russian you can go to Wikipedia. However, you will have to "fact check", i.e., you can't be sure of the quality or comprehensiveness of the information.
If you would like a quick way to get a list of sources with biographies on Shostakovich start with the World Biographical Information System produced by Saur. You can see the list of sources for Shostakovich at http://db.saur.de/WBIS/basicSearchResult.jsf . Note that the spelling used to obtain this list was Sostakovic.
Print or digitized encyclopedic sources are always a good starting point. Access to any of the "scholarly" encyclopedias, i.e., those with articles signed by the authors and that include bibliographies, is very useful. Some of these are online. However, most online encyclopedias are more basic. They are still useful for an overview. Dmitri Feofanov and Allan Ho's Biographical dictionary of Russian/Soviet composers. (New York: Greenwood Press, 1989)
If you are looking for Russian biographical materials you might start with http://biografia.ru . There are a number of Russian online sources for biographical material and a yandex.ru search for "шостакович биография " will yield a number of results.
If you are looking for something more detailed and in the context of music, you will want start with either the New Grove Dictionary of Music and Musicians (780.3 G91m2001) or the Soviet Музыкальная Энциклопедия (780.3 M9882). Both of these sources have online incarnations. The Grove is available online at the University of Illinois Library. If you are a student or otherwise affiliated you might want to take a look at the excellent, and extensive entry there ( http://www.grovemusic.com/index.html ). The Soviet encyclopedia is available on the web (http://www.music-dic.ru/). However, it is not the complete entry as it appeared in printed form. It lacks the very extensive bibliography offered in the printed edition.
If you are looking for extensive information on the life of the composer you will want to consult one of the bibliographic sources cited here or check the bibliographies in The Grove and/or the Музыкальная Энциклопедия. Both have excellent bibliographic sections that will lead you to a number of sources on Shostakovich.
There are also general biographical resources that can be very helpful such as the numerous composer websites, Russian and English, online links to media. PBS.org has some excellent material on Shostakovich including a disucssion of the quartets.
This Russian film traces the life and career of Dmitrii Shostakovich. There are a number of biograpical videos online for the composer so searching Google or YouTube could prove very fruitful.