Is there a word for "rule by criminals"? Not kleptocracy, which is rule by thieves, but more broadly by criminals.
Kleptocracy (or kleptarchy) does derive from the Greek word for “thief,” but the English word does not literally refer to thieves. Instead it describes a common form of government corruption which features various abuses of power including, but not limited to, embezzlement of public funds.
Mafia state is a related term for governments which have close ties to organized crime. Again, it does not literally refer to the Sicilian Mafia – a mafia state can be tied to any sort of crime syndicate.
You may be interested in kakistocracy, definition from Wiktionary:
Government under the control of a nation's worst or least-qualified citizens.
This is an interesting question because it depends upon your meaning:
If you mean the government is filled with people who are not adhering to the laws of that they enforce: this would just be a corrupt government, riddled with corruption, etc. etc.
If you mean the government is filled with people who've committed crimes, but are now functioning as a standard government that is a different story. The government would be whatever form it took. (e.g. A republic of thieves doesn't cease to be a republic.)
Carribean Pirate crews were well known for their democratic governance.
But, several terms have been used to describe "legitimate" governments whose members' morality was questionable. I've seen criminocracy used in the manner you're describing. (Also, thugocracy, and others.) For the most part, if you use the suffix -ocracy with a criminal sounding prefix, people will get your meaning.
Through-out history this is most often referred to as an oligarchy. Brad's definition is good too but I think this is the second time I have seen it. Oligarchy carries the connotation of criminal/military rule.
ol·i·gar·chy: a small group of people having control of a country, organization, or institution.
"the ruling oligarchy of military men around the president"