Which punctuation (if any) should accompany the inclusion of Japanese characters in an English text?

Since you are writing in English the ordinary punctuation rules for English would apply. Use square brackets [] or parentheses () to offset an entire parenthetical sojourn into kanji or kana, and punctuate as necessary.

Sometimes you don't need to set the words off at all, since written Japanese is so far from English that it sets itself off quite nicely. Example from a recent chat post of mine:

When Reagan was president he visited Japan and addressed the Diet. He tried to say the words 日米の友好は永遠です ("Japanese-American friendship is forever") in Japanese. What people listening on the radio heard, though, was 日米の湯女は大変です ("Japanese-American bathhouse prostitutes are a real handful"). Whereupon the wife of the person relating the story sang out "I wonder how he found that out so quickly. He's only been here for two days."

Notice that the English translation is the only part set off in parentheses. Had I wanted to include the transliteration as well, I'd have done something like this:

He tried to say the words 日米の友好は永遠です (nichibei no yukou wa eien desu, "Japanese-American friendship is forever") in Japanese.

Here I distinguish the transliteration from the translation with italics. I have seen transliterations rendered with furigana, however, the tiny kana over the kanji that give the pronunciation for Japanese speakers who may be unsure, but this would speak to a more advanced readership than you may be addressing. Also, furigana would be difficult to render in standard English writing software. I'm not even sure how you would do that, though obviously it must be possible.

All of this may be a matter of personal style, but personal style seems to be the norm. All of the English works I have read on Japanese handle such things in the ways that suit their needs, and these can be quite different. Sometimes kanji clips are parenthesized, sometimes rendered whole (as above) and sometimes they are even set off on their own lines in the middle of a paragraph. You really have to choose what works for you. I'd suggest consulting such works and choosing a method based on works that are closest to what you are trying to accomplish.