Vice President is usually capitalized when it is a title that comes immediately before the name of the vice president of a country—e.g., Vice President Biden. When it is a title that applies to other types of vice presidents (e.g., vice presidents of companies and universities), it is rarely capitalized in edited publications, but it is often capitalized in the official documents of companies, universities, etc. It is also capitalized when it is part of an official job title—e.g., Vice President of Investor Relations. When it doesn’t come before a name and is not part of an official title, it is not capitalized—e.g., They escorted the vice president to Air Force One.
Outside the U.S., vice-president is usually hyphenated in all its uses. In U.S. publications, it usually lacks the hyphen, though vice-presidential is hyphenated because it is a phrasal adjective. Incidentally, there’s no reason to capitalize vice-presidential.
None of these things are rules, however, and different writers and publications have different preferences. For example, it’s easy to find instances of uncapitalized official job titles, instances of vice president unhyphenated in Canadian, U.K., and Australian publications, and instances of vice president uncapitalized when it immediately precedes a name. If you need more straightforward guidance, check your preferred style book.
When it immediately precedes the name of the vice president of a country, vice president is always capitalized, but is unhyphenated in the U.S. and usually hyphenated everywhere else—for example:
On Wednesday, he heads to Virginia for three days of “debate camp” ahead of next week’s face-off against Vice President Biden in Danville, Ky. [Washington Post]
Members of the powerful politburo, including President Hu Jintao, Vice-President Xi Jinping and Premier Wen Jiabao, paid tribute to those who fought for the Communist republic [BBC]
Several security service buildings are located in the area, which also houses the office of Vice-President Faruq al-Shara. [Sydney Morning Herald]
It is also capitalized when it is part of an official job title—for example:
On Sept. 13 in a letter to students, Dean John Boyer and Vice President of Campus Life Karen Warren Coleman announced that they agree. [Chicago Tribune]
Vic Gundotra, Google’s Senior Vice President of Engineering has announced that Google+ has now surpassed the 400 million user mark since its public debut 12 months ago. [Independent]
Elsewhere, vice president is not capitalized—for example:
Alberta premier Alison Redford … has been less available to the media than a certain disappearing Chinese vice-president this summer. [National Post]
Senegalese lawmakers on Wednesday scrapped the Senate, or upper house, and the post of vice-president in a bid to save money for flood victims. [News24]