Take the vase to an antique dealer or specialist to determine its age. Japan has no native writing system but was introduced to the Chinese writing system in the sixth and seventh centuries. Over time, this writing system was simplified, then further simplified. Modern Japanese uses roughly 2,000 Chinese characters plus two syllabaries of 46 characters each. Knowing the estimated age of the vase will help determine the most likely style of writing to look for.
Scan or photograph the markings on the vase using a high-resolution scanner and avoiding using a flash, if possible. The clearer the images, the better for translation.
Inspect the markings closely in the scanned or photographed images. Often there are long trails from the calligraphy brush that may have implied words or may simply be the artist's style of writing.
Print several copies of the images, if desired, and trace over the calligraphy as you try to get a feel for what was written or implied. If you have no knowledge of Japanese writing systems, it might be best to take the images to a Japanese language professor or historian.
Write down possible characters you recognize from the writings on the vase. Use these characters to search for words in a Kanji (Chinese character) dictionary. Sometimes figuring out the main characters gives the best general interpretation of what the artist wished to convey.
Consult with a Japanese language professor or historian regarding the writing on the vase, if desired. Writing on artwork and pottery may be a famous poem or a poem compiled by the artist. A Japanese language or literature professor is likely to recognize this type of writing and be able explain the meaning and translation.