This depends on your definition for “HTML5”. HTML5 is work in progress, and we normally see a new version literally every day, if we regard HTML 5.1 Nightly as the current definition for HTML5, as many people do.
The quickest way is probably anyway to submit the document to http://validator.nu in HTML5 mode and then fix whatever problems are reported. This does not guarantee conformance to HTML5 (which here means “what the authors of the validator regard as applicable in HTML 5.1 Nightly”), however. Many conformance requirements can only be checked by a human being that knows HTML5 and understands the content of the page.
It might not be very quick, depending on the page. You might need to spend quite some time in learning HTML5 rules and also CSS, since many constructs that are valid in XHTML 1.x, especially in XHTML 1.0 Transitional, are forbidden in HTML5 (though they work excellently in browsers) and need to be replaced by CSS (mostly).