The Klingon Empire is ruled by a Chancellor with the support of the Klingon High Council. If a Chancellor dies of natural causes, his replacement is chosen by an Arbiter of Succession, who chooses from eligible candidates (usually Council members themselves) in a process called the Rite of Ascension, based on rules established in Klingon law. A Chancellor with some foresight can decide who will be the Arbiter of Succession after his own death. On the other hand, if the reigning Chancellor is killed by another Klingon, then the killer has the right to claim the Chancellor's position, or to appoint a new one (unless the killer is unidentified, in which case the Rite of Ascension takes place). Shockingly, a Klingon's combat skills seem to be the main method of determining their suitability of the the Chancellor's role, rather than any political skills.
A clone of the Empire's founder, Kahless, was recently crowned Emperor, but his title is ceremonial and carries no real governmental authority.
The Klingon Empire controls a large volume of space that shares borders with the Romulan Star Empire and the United Federation of Planets. The full extent of their territory is unknown, but it is probably not substantially greater than that of either of its neighbors.
The Klingons are one of the most formidable military powers in the Alpha Quadrant. Martial traditions are so widely revered in Klingon society that almost every Klingon has at least basic military training. Similarly, virtually every Klingon has personal weapons and body armor, and attaining any sort of political influence requires a good military service record. Klingon technology is comparable to that of the other major Alpha Quadrant powers, and they invest heavily in military assets.
In an alternate timeline, the Klingon Empire was shown to be decisively winning a war against the United Federation of Planets (TNG, "Yesterday's Enterprise").