Beyond trade and mission work, European powers weren't really active in Japan during the Sengoku period, but that's not to say that the first two categories didn't have their own national interests or that they didn't keep a close eye on the political conflicts of this period. The political shifts of the Sengoku period dictated the shifting status of Christianity in Japan, which was of direct interest to one major group of European observers in Japan, the Jesuits, and of indirect interest to European traders who might benefit from the rise or fall of European visitors' freedoms and privileges. Italian merchant Francesco Carletti gave a description of 16th century Japan in his Chronicle and gave the following description of Toyotomi Hideyoshi --

At the time when I reached that kingdom, the universal overlord was Taico Sama, who also was called Quam, Bacco, Dono, and – earlier, when he had been a soldier and a private – Fashiba. Later he became the tyrant of all that region, though he was not born king or even of royal blood.

Carletti also documents Hideyoshi's 1597 crucifixion of twenty-six Christians, giving a detailed description of the method of execution and display used in this. In general, European commentators are keenly interested in the political structures of Japan -- hierarchies, titles, feudal obligations, crime and punishment, and so on, both in comparison with European politics and on its own merits -- but due to the position of European observers, they can offer no military backing to any party and they largely stay out of direct conflict in the interest of preserving trade interests. These assorted personal accounts by Europeans (largely Portuguese and Italian) are what give Europeans outside Japan their image of the political situation in Japan.

Jesuit commentators describe engagement with Sengoku-era politics, usually with respect to their continued proselytization efforts. There's a mix of observation and partisan support. Jesuits themselves don't constitute much of a political force in Sengoku terms -- they can only hope to influence the political forces of Christian daimyo by prevailing on their religious sympathies, so, unreliably and indirectly -- but the political force they do exert is enough to focus anti-Christian political pressures on the Jesuits themselves along with less numerous Catholic orders like Franciscans.

Jesuits reported back extensively on Japanese politics to their confreres in Goa via letter, and you can gain a picture of their shifting expectations on these civil struggles -- both hope that the success of Christian daimyo or otherwise Christian-friendly politicians might alleviate hostilities toward Japanese Christians, and apprehensions that persecutions might increase. The biggest Jesuit concern was continuing access to Christian daimyo, and the biggest area of conflict with Hideyoshi et al. was perceived inordinate Jesuit influence over Christian daimyo, as a roadblock to Japanese unification and an impediment to Hideyoshi's own goals. After the death of Hideyoshi, Jesuit commentators seem to have backed Tokugawa Ieyasu for his perceived amenability toward Christianity and their belief he'd be the likely victor of the current succession conflict. Jesuit accounts trace Ieyasu's actions and accomplishments quite closely, both as general historical observations and in terms of their effect on the state of Japanese Christians. I talk a little about the Jesuits and Tokugawa Ieyasu in this earlier thread.

Dutch commentators post-1603 provide quite a bit of commentary on Japanese politics and political actors, usually in the context of Dutch delegations trying to negotiate their unique status in Japan after the expulsion of all other Europeans, but Jesuit political accounts more or less stop with their expulsion from Japan and all continuing accounts (by European and Japanese Jesuits) focus on politics only with respect to the persecution of Christians.

Some sources:

• "16th Century Japan and Macau Described by Francesco Carletti (1573-1636)", Elisabetta Colla

• "Tokugawa Ieyasu and the Christian Daimyo during the Crisis Of 1600", João Paulo Costa Oliveira e

• For European accounts of Japanese politics during the Sengoku period and beyond, Volume III of Asia in the Making of Europe, by Donald F. Lach and Edwin J. Van Kley