Tokyo's official status as capital of Japan is generally not in dispute, but it is not legally defined. In fact, there is a dispute as to exactly when Tokyo became the capital: some say that it occurred when Tokyo prefecture was established in 1868, others say that it occurred when Edo Castle became Tokyo Castle that same year, and still others say that it occurred when Tokyo Castle became the Imperial Castle (now the Kokyo) in 1869. Historically speaking, while there was an Imperial edict transferring the capital to Heiankyo, such a basis has never been provided for the transfer from Kyoto to Tokyo. So, today, there are some people who say that since the transfer to Heiankyo was valid, Kyoto is still the capital of Japan, while some say that Tokyo and Kyoto are both simultaneously capitals of Japan.  (http://ja.wikipedia.org/wiki/%E6%97%A5%E6%9C%AC)
Conceptions of the capital of Japan before the Meiji restoration
During the Edo Era, Japan was often said to have three capitals, or miyako (都). Edo was the capital of the Tokugawa shogunate, Kyoto was the residence of the Emperor of Japan (therefore making it the capital of culture and tradition), and Osaka was the unofficial capital of the merchants.  (http://www.mlit.go.jp/kokudokeikaku/daishu/online/lec16.html#b)
Arguments in favor of Tokyo as capital
The location of the Diet
After World War II, the new Constitution of Japan transferred the state's sovereignty from the Emperor to the people, as represented by the Diet of Japan. The popular logic then became that the site of the Diet denoted the capital of Japan.  (http://ja.wikipedia.org/wiki/%E6%9D%B1%E4%BA%AC) This is the most concrete basis for legally recognizing Tokyo as the sole capital of Japan, since the Emperor has no governing power and all other state institutions are based in Tokyo. However, it falls short of an explicit statement that Tokyo is the capital.
The "capital area" in Japanese law
While no laws have designated Tokyo as the Japanese capital, many laws have defined a "capital area" (首都圏 shutoken) that incorporates Tokyo. Article 2 of the Capital Area Consolidation Law (首都圏整備法) of 1956 states that "In this Act, the term 'capital area' shall denote a broad region comprising both the territory of Tokyo Metropolis as well as outlying regions designated by cabinet order." This clearly implies that the government has designated Tokyo as the capital of Japan, although (again) it is not explicitly stated, and the definition of the "capital area" is purposely restricted to the terms of that specific law.  (http://law.e-gov.go.jp/htmldata/S31/S31HO083.html)
Other laws referring to this "capital area" include the Capital Expressway Public Corporation Law (首都高速道路公団法)  (http://law.e-gov.go.jp/htmldata/S34/S34HO133.html) and the Capital Area Greenbelt Preservation Law (首都圏近郊緑地保全法)  (http://law.e-gov.go.jp/htmldata/S41/S41HO101.html).
It is interesting to note, however, that this term for capital was never used to refer to Kyoto. Indeed, shuto came into use during the 1860's as a gloss of the English term "capital."  (http://www.mlit.go.jp/kokudokeikaku/daishu/online/lec16.html#a)
Official government positions
In 1941, the Ministry of Education published a book called "History of the Restoration," which is still used by modern scholars. This book referred to the "designation of Tokyo as capital" (東京奠都 Tōkyō-tento) without talking about "moving the capital to Tokyo" (東京遷都 Tōkyō-sento). A contemporary history textbook states that the Meiji government "moved the capital (shuto) from Kyoto to Tokyo" without using the sento term.  (http://www.mlit.go.jp/kokudokeikaku/daishu/online/lec16.html#a)
Recently, there is a movement for transferring the capital from Tokyo, and the Gifu-Aichi region, the Mie-Kio region and other regions bid for it. Officially, the relocation is referred to as "capital functions relocation" instead of "capital relocation," or as "relocation of the Diet and other organizations."  (http://www.mlit.go.jp/kokudokeikaku/daishu/English/english.htm)