I second what @Fumiko and @CherryBunny said, and may I recommend Atsuhime? The Tokugawa weren't royals per se, but I hope Ooku and Atsuhime aren't too far off from what you're looking for (life in a cloistered place, inner-palace intrigue, that kind of stuff).
Happy to see this was brought up, I was kinda curious too why don't we see shows about japanese monarchy. I've been watching japanese dramas for years and I can't remember any shows about japanese monarchy.
I'm wondering if it has something to do with the way the imperial family was seen through history ( supposed divine status, or currently, of divine descent) : is it some kind of taboo?
Historically speaking, it does'nt seem that imperial family members really played a concrete and big role in building the Japanese state (at least obviously not during the Sengoku and shogunate eras). The current statuts of the emperor does not give him a big role, and he is more a moral reference or a point of stability in the moving political world than anything else. (my point of view)
@sumithecat has the exactly right idea! The Japanese royal family are part of a "divine lineage" with the emperor as a direct descendant of the sun goddess. Because of this mythical prehistory, prying too closely or questioning too much is very taboo. In fact, very little information is known about the royal family and they exist behind what many journalists describe as a "chrysanthemum curtain" that is almost completely impenetrable. All media surrounding them is heavily heavily censored, and they very rarely do interviews (and the interviews they do hold are all scripted by the household agency). They also have no passports, no money in their name, no processions in their name, and no last name either. The imperial household agency wouldn't even tell the press what breed of dogs the royal family had for like, five years. Also gonna be honest and say there's still quite a bit of controversy surrounding the royal family re: WWII war crimes, right-wing nationalism etc.
Long story short I don't think a drama would ever get cleared!
This is an interesting topic and i believe many of us (including me previously) have thought the Japanese system would have been similar (or at least have some similarity) to the general monarchy system in the rest of the world., but that was a misconception.
While some of you (@Hanna @Sumi) might have the point about the taboo and the sensitivity of this matter, in my opinion (from what i read on wiki and observe from watching historical drama), the lack of story about Japan imperial is simply because there's nothing to tell about! Like what @Sumi mentioned, they did not play big role in building the state.
Japan always have the King even until today (the King never been thrown out), but the King have lost power since year 1000 AD and only got that back during Meiji Restoration just before WW1. You see how long was that? And that was not last long either.
During those time, the one with most power was Shogun, and then losing to the Oda and to Toyotomi and then eventually the slick Tokugawa tricked everyone and become a new version of Shogun with more power than the traditional one. This last until Meiji Restoration.
So, to see more involvement of Japanese royal you have to watch drama/movie before the shogun era (i.e before year 1000). One that come to mind is Taira no Kiyomori. Here you can see how corrupt the palace was. Keyword: Incest.