In Sengoku you control a Japanese noble in the Sengoku Jidai period. You develop provinces, hire and direct armies, conduct diplomacy, and once your noble inevitably dies, you continue playing as his heir. This continues until someone manages to unite half of Japan for 3 years, thereby becoming shogun.
The game is very similar to the next game by Paradox Development, Crusader Kings 2, The main difference besides the setting being that whereas CK2 is (arguably) primarily about the relations and personalities of characters, Sengoku is primarily about waging war.
Declaring war in Sengoku is as simple as spending some honor. (You need no casus belli.) However, you don't want your honor to fall too much, or your subordinates will cause you trouble, so you need to replenish honor by making gifts of money and land, acquiring titles at the imperial court, build temples, etc.
Unfortunately, while this honor mechanic is fun, it doesn't really keep you occupied very much, and unlike CK2, it's the only non-war element of the game with any real substance to it. There's very little depth to diplomacy, plotting, dynasty-building, or ambition-chasing; the economic model is very simple; and there is no tech-tree. At any time that I wasn't at war, I set the game to maximum speed until I had enough resources to start another war. There simply wasn't much of anything to do.
Unfortunately, combat isn't any better than in CK2 either. It's mostly a matter of creating a big stack with a decent leader and sending it to the enemy. You get to decide where the armies go, but once you meet the enemy and combat starts, it's all purely abstract with some numbers going down over time. It's more fun than it probably sounds, but not a lot more. And again, CK2 does it better, since there you can at least divide your troops over the flanks and center, assign leaders, and utilize a greater variety of troop types.
In conclusion, while I can't say that Sengoku is a bad game, I simply cannot recommend it, since it's been so thoroughly obsoleted by CK2.