Carthage had been a monarchy originally, but by the third century BC the senior executive officers of the state were two annually elected suffetes. It is not known wether this office replaced the monarchy or developed from it, but the greek use of the word basileus which means king makes it possible that it is the former. It's debated what kind of nature the Punic monarchy had but it's possible that it was an elective office. Wealth and merit was important in the election of the suffetes, but unlike their Roman counterparts they did not hold military command; they had instead supreme religious and civil power. A council of thirty elders (Gerousia) acted in an advisory capacity and was supervised and possibly elected from another body the Council of 104. If the suffetes and the Elders agreed something then they had the power to implement it. If they failed to reach an agreement, then the proposals were taken before the Assembly of the people to decide on the matter (here any citizen was permitted to make a counter-proposal). It is clear that a relatively small number of noble families dominated the council and probably controlled the offices of the Suffetes.

Much of the details of the internal politics of the city of Carthage however is not known. Some of the writings may point towards factionalism and internal disputes but it would be impossible to describe thes with precision. Aristotle praised Carthage's constitution for balancing monarchy, aristocracy and democracy, which allowed it to avoid internal instability that many of the Greek states suffered from.

Carthage seems to have been rather stable when it comes to politics, although it seems to have been a system dominated by the wealthy and most of all they benefited the most from it.

Source: The Fall of Carthage P. 30, Adrian Goldsworthy.