You should study Banana Republics
In political science, the term banana republic describes a politically unstable country with an economy dependent upon the exportation of a limited-resource product, such as bananas or minerals. In 1901, the American author O. Henry coined the term to describe Honduras and neighbouring countries under economic exploitation by U.S. corporations, such as the United Fruit Company. Typically, a banana republic has a society of extremely stratified social classes, usually a large impoverished working class and a ruling class plutocracy, composed of the business, political, and military elites of that society. The ruling class controls the primary sector of the economy by way of the exploitation of labour; thus, the term banana republic is a pejorative descriptor for a servile dictatorship that abets and supports, for kickbacks, the exploitation of large-scale plantation agriculture, especially banana cultivation.
A banana republic is a country with an economy of state capitalism, whereby the country is operated as a private commercial enterprise for the exclusive profit of the ruling class. Such exploitation is enabled by collusion between the state and favored economic monopolies, in which the profit, derived from the private exploitation of public lands, is private property, while the debts incurred thereby are the financial responsibility of the public treasury. Such an imbalanced economy remains limited by the uneven economic development of town and country, and usually reduces the national currency into devalued banknotes (paper money), rendering the country ineligible for international development credit. (Source)
Banana Republics are unstable because...
A rigid class system guarantees the labor force is always impoverished.
The elite ruling class includes political and military leaders who are kept in power by external businesses. In the case that a political/military leader happens to stage a coup and take control, they cannot maintain that control without the benefit of external businesses.
External businesses are required (OK, preferred) because they never have a ruling interest in the country nor have incentive to humanize the labor force. Unlike the governments of other countries, which an be influenced more by their own populations, businesses are frequently capable of acting without any kind of independent oversight.
NOTE #1
While there are still many instances of Banana Republics in the world today (business will always try to capitalize on cheap labor), the more technologically advanced the world (think "fast communication and cheap pictures/video"), the easier it is for grass-roots activists to enact change that limits the influence of business. So, while Banana Republics still exist today, it's nothing like it was in their heyday of the 1900s.
NOTE #2
It is possible for governments to do this, although Hollywood makes it look simpler than it really is. It's almost a trope in itself that the CIA is constantly forming, supporting, and dismantling 3rd-world governments. However, while I strongly suspect that this isn't as ubiquitous as any James Bond movie would suggest, it has happened. Consider Manual Noriega of Panama or the Marcos' of the Philippines, both of which were supported by the U.S. until they were overthrown. Similar stories could be told about Asian countries in the 60s-70s (Cambodia, Korea, Vietnam, etc...) and the Cold War era.
The point here is that the overlord (usually a business) doesn't really care who's in power so long as their interests are protected — and will often provoke changes in the government themselves to ensure their interests are protected. From the perspective of a Banana Republic, there's nothing more inconvenient than a leader who gets too greedy or grows too much of a conscience.
And if you want to get a better handle on this while having a good time, track down a copy of the game "Junta," which will do a great job of teaching you how the instability works while letting you experience your inner dictator.