It Depends on the Country in Question
More on why your solution depend son the country you want to destabilize in a minute. Before we get into that, I want to make two key points:
Point 1: Force is Out
You will not take down any major government with force. Dirty bombs/terrorism/etc. will not take out any major government - it will, if anything, have the opposite effect. Look at France as a recent example - after terrorist attacks, police power was significantly increased with the full support of a majority of its citizens. When attacked, people dig deeper to protect themselves.
Point 2: Conspiracies Rarely Work Out
Plotting with other people rarely works out. Take a look at this research to see how quickly conspiracies unravel as more people are brought in. So big plots involving 50+ people trying to blow things up or assassinate people are out. You should plan on having to do everything yourself, which likely removes assassinations and any other 'wet work'.
Solution A: Get Yourself Elected (see requirements)
Time Frame: ~5-8 Years
This solution requires that you are a native citizen of the country you wish to take over. Foreigners may be allowed to run in the federal elections of whatever country you are targeting, but you will simply never win. Also the target country can't be China or any other nation that does not have real elections, for obvious reasons.
With 1Tn in reserves this should be trivially simple. The US elections are the most expensive in the world, but the record spent is still just over 1Bn, a mere thousandth of what you can bring to bear.
In advance, you will probably want to buy a number of media companies ahead of time (another few $Bn), then hire a TEAM of the best political planners and you're off and running. The team thing is important, because at the end of the day there are only maybe a dozen or so people that have experience with managing national campaigns... and you could hire ALL of them. That not only gives you a better knowledge pool, it also denies your rivals access to competent management.
At the same time, be sure to start electing cronies to all state/provincial/regional (depending on country) elections. Somewhat slimy people with no real backbone of their own who will just do whatever you say so that you will fund their re-election campaigns in the next cycle as well. Competence is not needed or desired.
Once you have been elected, start making subtle decisions that undermine public trust in other institutions. For example, start selling off significant public assets (ostensibly to pay back national debt, which nearly every country has) - this will reduce the resiliency of the federal system.
Then start a very negative PR campaign that pictures you as the leader that is finally fixing things (an idea that your media outlets are keen to promote). This is where things really stop working, because a government can only function when its citizens believe in it. For example, in the USA one might start calling the IRS a bad idea, state that you have uncovered gross fraud, and do the same with Homeland Security. Then do something nuts like encourage people not to pay their taxes - I bet it would work pretty well. It's obviously harmful to the country, but you have filled any balancing seats of power with your cronies, so you won't be impeached.
At the same time, disband significant portions of the military. Your cover will be that you are doing it because of the "unacceptable military industrial complex" and "people who are making billions of dollars off of it". You won't be required to give any hard proof. But the real kicker here is that when you do this, you fire every general (and on down) that isn't a complete crony. Then you again hire/promote cronies. At this point you have replaced the actual military with one totally under your control.
Next raise taxes on corporations radically - after all, they "don't support the 99%!" - and raise tariffs on goods imported and exported significantly. The economy will begin to crumble very quickly.
At this point you have a government that people don't trust, rampant economic crisis, and a military that you control fully. You ought to be able to affect any type of government change (including anarchy) that you desire.
Solution B: Do What You Can
Time Frame: 8-15 Years
Let's say you can't be elected in the country you want to destabilize - a much harder situation. Using force is out, because people will just entrench (unless you're trying to destabilize a very small country, which is another story). You can't be elected, so you can't make the decisions yourself.
But you can still buy ALL the media outlets active in the country. And you fill them with investigative journalists who probe all areas of the government. You also hire detective agencies to dig up dirt for them that the journalists can't/won't get themselves. This creates a years-long tirade in all the news about how the government can't be trusted. And yes, that would be the case in even the best-run country, because all an article needs to find is one bad worker in the millions of public workers to create a 1-week news cycle... I'm sure you can find a few hundred easily. This undermines public confidence.
As confidence erodes with the main parties, throw your financial support behind the "crazy party". The one that no-one would have voted for when they believed in the government. In the USA that might be Libertarians or Communists. Extreme right- and left-wing parties exist everywhere. Recent elections should show that in many European countries if those parties were better funded they could have won elections in many countries. Plus your detectives are digging up dirt on the other guys :)
Next you need to try to demolish major industries. If the country is heavy into tourism, find out why and attack it. As an example.... have your media outlets around the world begin to publish news of every piece of violence that occurs in the country, and call the political situation "unstable" to discourage visitors. You can also buy transportation companies used by tourists and run them into the ground. Find landmark hotels and fire 95% of the cleaning staff. Let people come in, but make sure they have nothing good to say about their visit. The industry and its feeding businesses will collapse within a few years.