I fantasize about doing that sometimes, but usually, you can't get into politics without being a citizen — either due to legislation or because you won't be accepted as a representative by the population, for obvious reasons.
But assuming you can get into politics, you'll still have to deal with an enormous struggle for power (be it elections you have to win or powerful people you have to convince) in order to get into a position to even make any change.
Thus, this endeavour is practically impossible — it's already difficult enough to change something in your fatherland.
AFTER tokenadult's EDIT: Your edit clarifies a lot. But fraternalism is still difficult to achieve: if you are from a different race than the native population, you'll have a hard time being accepted as their representative.
An interesting example for this might be Singapore, where Lee Kuan Yew pushed the country from third to first world in 50 years. Although he is Singaporean, he brought in a lot of foreign knowledge as he had studied abroad. And the same issues with corruption/illegal methods some commenters mention here can be observed: in the early days of his career, he arrested his opponents, the communists, without any legal process to prevent them from jeopardizing his plans. Later, he made some radical decisions, that were not widely accepted by the population, bu that led to better results for the country (for example, he tore down old villages to replace them by high-rise buildings).
If you create a business that solves a local problem (local to the poor country of choice) good enough, and fairly enough, you can create ripple effects. Maybe not at national levels, but at regional and municipal levels.
Not easy, but not impossible.
(At least I hope so when I got my rose-tinted glasses on.)
Corruption. Remember that if you go to one of the 20 poorest or tyrannical countries of the word, you will probably go to one of the 20 more corrupt countries in the word. So while creating a successful business you will be "asked" to share a part of it, and I hope you can keep enough to create the ripple effects.
It's probably doable, but not without a whole bag of dirty tricks, and a group of well trained supporters... but you'd end up being like a secret agency manipulating foreign countries' politics, nothing new here, and if you were to succeed you'd give in to the temptation of using your "great benefactor, savior from poverty..." power for more one-sided goals.
If you want to change a foreign country for the better and not make a bloody (literally) mess of it, stay away from politics (ok, bribe who you need to to keep you humanitarian organization afloat if needed) and work to improve education, basic healthcare and economic growth by means of bringing foreign investments to businesses in a poor country or things like that. Any other way to do it and you'll probably end up on "the dark side" sooner or later :)
It would be amazing to have an entity, sort of like the US military crossed with the Red Cross, which just went into conflict zones or repressive regimes to provide infrastructure of government (power, water, education, transportation, law enforcement, security for civilians, etc.). Capable of defending itself, but not taking sides other than providing that infrastructure to everyone. Nominally this is the UN's mission in some countries, but it doesn't really do it.
Satellite TV (Al Jazeera and western sitcoms) largely accomplished this for information in the greater Islamic world. Doing the same for non-information goods would require actual infrastructure on the ground, though.
There are very few things that people in power hate more than someone else succeeding at something without consulting them first. I'm afraid you organization would need to be almost all military.
Most of these countries have fairly ineffective militaries, though, so even a lightly armed force could deter them. Blackwater and at least 2 other PMCs in Iraq could hold their own vs. any failed-state force or militia. It wouldn't need to be like the US Army or USAF fielding high-end weapons where a huge tail to tooth ratio came into play.
Do you know any example of a humanitarian foundation contracting services from Blackwater/Academi type PMCs? Sounds like a workable combo that hasn't been tried before. Though you'd have to have the support of the government even if you can defend yourself from guerrillas/militias and thieves, but by simply receiving this support in a messed up country with "alternative governments" fighting for power, you'd end up taking sides, loosing your neutrality and becoming vulnerable to all sorts of bloody acts, even if they are directed against the people you help and not against yourself - imagine you're helping a school and a bunch of guys from a local guerrilla fighting for power show up and take the school children hostages and tell you they'll kill them if you don't pack your gear and leave the area: even if you have the resources to kill the untrained kidnappers and save all children, by doing so you seriously piss up the fraction that sent those guys and open things up to other alike situations that will not all end well. Something like a PMC force would only be usable against thieves / poachers / drug-dealers... any other "defensive" use would be "taking side" and pull you out of the neutrality zone messing up everything.
To actually be able to use a PMC for such situations, you'd need the kind of force that could win a fight/kidnapping situation by using non-lethal force against lethal force, because using lethal force will put you on one side or another and compromise everything. And to be able to reliably win a non-lethal vs. lethal force situation you'd basically need your PMC to be more advanced and well trained than the US military (!!), at least on the non-lethal weaponry and people manipulation parts... and at that point something like the US (or other "big guys" with interest in the region) would perceive you as a potential treat to them, despite your humanitarian mission, bringing up whole new levels of "fun" :)
Sorry for the pessimism, but these is why I don't like touching this kinds of problems, as I can only find an infinite amount of reasons of why things wouldn't work.
Plenty of NGOs and UN entities hire either local security or external PMCs to provide security. They usually can't afford top end like Blackwater, though.
It would be easier to stick to services like running an airport (eg Mogadishu kept operating...), water, hospitals, power, etc. but another option would be to run "refugee/IDP centers" like the UN allegedly was doing in the Balkans (where they failed spectacularly). If people live inside a camp they are relatively easy to protect.
The point behind lethal force by a PMC is to have overwhelming force. Blackwater had an Air Force, and essentially could call in favors from the CIA and DOD as needed. Outside of the 2004 Fallujah incident, they never were in serious trouble, and after 2004, the USMC basically tore the city apart, largely over the bridge incident.
There would be no way to do this kind of thing without US support (tacit or otherwise) and at least non objection by regional powers, but if you stuck to "help IDPs and execute on international obligations" you would get UN, USAID, etc funding. The issue is that most people in NGOs tend to be bleeding hearts, and also humanities people vs infrastructure people, so thy don't have these capabilities in house; they have to contract them in. The exception are medical groups -- there are huge numbers of great medical facilities run by religious and other charities, and their neutrality is generally respected.
Even after they renamed the first time (to Xe), all the guys continued wearing Blackwater caps and shirts. Even the nominally-separate Presidential Airways guys were wearing Blackwater clothing with the bear claw logo :)
See my edit regarding the "dark side". You are generally right, but there is at least one example that turned out well despite occasionally taking advantage of illicit maneuvers.
But assuming you can get into politics, you'll still have to deal with an enormous struggle for power (be it elections you have to win or powerful people you have to convince) in order to get into a position to even make any change.
Thus, this endeavour is practically impossible — it's already difficult enough to change something in your fatherland.
AFTER tokenadult's EDIT: Your edit clarifies a lot. But fraternalism is still difficult to achieve: if you are from a different race than the native population, you'll have a hard time being accepted as their representative.
An interesting example for this might be Singapore, where Lee Kuan Yew pushed the country from third to first world in 50 years. Although he is Singaporean, he brought in a lot of foreign knowledge as he had studied abroad. And the same issues with corruption/illegal methods some commenters mention here can be observed: in the early days of his career, he arrested his opponents, the communists, without any legal process to prevent them from jeopardizing his plans. Later, he made some radical decisions, that were not widely accepted by the population, bu that led to better results for the country (for example, he tore down old villages to replace them by high-rise buildings).