Step 1: Laying the Foundations
As mentioned by everyone, the key obstacle in invading and establishing control of the United States is its sheer size, a factor that is generally a historical problem. Look no further than World War II, when the Soviet Union was invaded by the Reich, but was saved by its size. Factories could be evacuated to the Ural mountains, troops redeployed from Asia, the vast spaces used to maneuver armies... This size is the biggest problem.
As such, the first step in preparing for an invasion would be to cut the United States down to size and weaken its economic, military, and political position. I believe there are several ways to tackle this, in no particular order:
* Overseas Military Campaigns
As shown by the most recent wars carried out by the United States, foreign invasions and occupation are extremely costly and put a strain on the economy at home. One of the ways to weaken the U.S.' position is to provoke them into another invasion of a foreign country, provoking a long drawn-out war. Several choices come to mind, though for now, Iran or North Korea would be good candidates.
Since attacks on U.S. targets in North America provoke the biggest reaction, a terrorist attack on a city in North America (such as Boston, San Francisco, Los Angeles etc.) with a trail of breadcrumbs leading to either of the two countries could provoke a large response. Of course, this would require fooling the various national security agencies. The method of attack would be the least of the problems, though given the nature of domestic security, attack methods that would use materials available in the United States and cause mass casualties at the same time would be preferable.
The other caveat is that the regions attacked by the United States would quickly destabilize. The obvious result of provoking the U.S. into attacking North Korea would be that Seoul would cease to exist overnight and the Armistice would fall apart, leading to further casualties. If Iran was tackled, or any other country in the region, Israel would likely be facing invasions of its own, as a natural ally of the United States.
* Economic problems
Another method would be to target the foundations of the economy in the United States. I'm not an economist, so my input on this matter will be limited, but the primary problem with triggering economic meltdowns (I use the term loosely) is the fact that any economic damage to the United States would drastically affect the rest of the world, especially countries relying on U.S. commerce.
* Internal problems
I feel like the best way to start weakening the United States would be to exploit pre-existing problems in the U.S., most notably the broad variety of THE GUV'MINT WANT TO TAKE MAH GUNS nutjobs and the various variations on the theme. Supporting these groups would provide a steady level of disturbance on the domestic front, but since they are few and far in between, with most Americans consenting to political apathy (something common to democracies, as far as I'm concerned), utilizing them to full potential would require more drastic steps.
Enter domestic terrorism. Encouraging and arming the most radical groupings and encouraging already unstable individuals can lead to a rise in terror attacks carried out against Americans by Americans. If the process is not detected and the groups are not defused by national security agencies, this will definitely lead to tightening domestic security and impose further restrictions on top of those already existing.
In general, I don't think U.S. citizens will care about increased intensity of security, as long as it doesn't touch the things they actually care about (as the passage of the PATRIOT Act shows, actual liberties rank pretty low on that list). Given the prevalence of gun culture and the prominence of gun-related discourse, targeting this aspect of American culture can yield potentially the most results. Of course, getting the federal government to act against its own best interests can be hard.
In my opinion, they should be forced to enact measures that have been long decried by various more or less insane groups and individuals, like a national registry of firearm owners or mandatory background checks. I'd have to think of it more, but ensuring that weapons carried out in the attacks (especially mass shootings) were acquired legally and did not end up in the hands of the perpetrator through unlawful means.
The obvious immediate result would be the equivalent of a shitstorm perpetrated by the right wing and likely the immediate radicalization of the more unstable gun owners and associations. I can't think of a single more destructive event than this.
* Balkanization
Obviously, the most effective weapon of weakening the United States would be to break it apart. It would be hard, but would definitely yield ample benefits, as smaller countries that broke off from the Union would not only become much more manageable and easy to influence, but would further destabilize the continent by the virtue of their existence.
The caveat is that for this to happen, the U.S. has to be already weakened by previous steps. In order for it to truly fall apart, though, fully fledged, legitimate separatist movements would have to be present, strong, and supported by the populace. Funneling funds and knowledge to them can only work so far.
The key, in my opinion, would be to reduce the standing of the federal government to the point where people would start believing they're better off on their own. And by that I don't mean fringe lunatics convinced the GUV'MINT WANTS TO TAKE MAH GUNS, but your average Joes and Janes, no longer possessing any loyalty to the Union.
Perhaps it would be best to start with states that have natural outwards tendencies. Texas is the usual choice in fiction, but would it actually work? Requires more research.
The added benefit is that fracturing of the United States automatically cuts apart its military and nuclear potential. And with said nuclear potential breaking free of the federal government (unless they decide on desperate military action, furthering the goals), I believe foreign powers would be interested in making sure it doesn't fall into the wrong hands.
* Overarching goals
I think it's good to keep your eyes on the ultimate goal all the time. In this case, it is the preparation of the American populace for invasion and conditioning them to see it not as an actual invasion, but welcome relief and help. It is true that strong patriotism is usually the norm, sometimes bordering on jingoism, but if circumstances could be created in which basic necessities aren't readily available and the quality of life drops sharply, I wager loyalty to the state would fly out the window. Especially if the rule of law and effectiveness of law enforcement agencies can be curbed to the point that safety isn't guaranteed.
In short, the United States needs to fail as a state, if not in practice, then at minimum in the eyes of its citizens.
