This would happen post-WW2 (probably late 50's-early 60's).
An activist thinks things in the USA have to change but knows (or is firmly convinced) that as long as only the Republican or Democratic parties are the only de facto options for the President no one who would actually threaten the status-quo for the wealthy elite could ever gain the power to do so.
What would he need to do to reach presidency as an independent despite the system (for that matter, is there anything he could do)?
The First Past The Post voting system tends to stratify into a two party (with occasional smaller parties) system. The classic case being
I really don't want him to get in, so I'm going to vote for the only other viable option as a vote for a 3rd party candidate would be a waste of my vote.
This is the real key to your problem. The electorate have to be sure your third party candidate is going to win, and that there isn't a risk of the vote going to the person they really don't want, over the person that they consider acceptable. Hence they almost always vote for one of the big two. The other source of 3rd party votes is the protest vote, but that tends to remain an insignificant minority.
If you want to encourage more parties into your system use an AV/IRV voting system. This makes it easier for small party candidates to get in as it allows tactical voting as second option rather than only option.
I'd quite like her to get in, so she's my first choice even though she's from a minor party. He's ok and a major party candidate so he's my second choice. That one is crazy so I won't put him down at all.
Tactical voting still leaves the 3rd party candidate a chance of winning which it wouldn't under FPTP.
If you want to guarantee more parties in your system use Party List Proportional Representation. In this case the lower your required threshold to get a seat, the more parties you'll get (and the crazier they'll be). You could also get a bad case of Belgium spending a year without a government because they couldn't agree a coalition, but at least you broke the two party system.
It's of course important to note that the two major parties in a two party state are not going to support anything that breaks their hold on power. That would be a classic case of the turkeys voting for xmas.
There is something called Duverger's law which suggests that the US political system causes a two-party system. So if you do not want to replace the current two-party system with a new one, you would have to change the Constitution. That requires the consent of lots of people who were elected under a two-party system. How about:
Today, Brazil has 35 registered parties, and there are some groups which are trying to create new ones. No party has, per se, any majority anywhere, so they are always creating, destroying and changing party coalitions. Over the years, parties were born, fused together, divided into smaller parties, extinguished and refunded, etc. So, it is always a changing situation.
The result of having so many parties is, of course, a big mess in the brazilian politics. What caused that was some democratic reforms that the weak and disorganized, although authoritarian and dictatorial, governement were forced to perform in the early 1980's. There was two parties during the dictatorship, a situation party (ARENA) and an opposition party (MDB). The opposition party were starting to get dangerously powerful and annoying in the vision of the situation party, despite the dictatorship persecution. The opposition was fierce and the dictatorship was unpopular and getting weaker and weaker.
So, the government decided to perform a major political reform. The purpose was to divide the opposition into many small competing parties in order to avoid that any of them would get any significant representativity to pose any danger to the situation party hegemony, and then make them fight against each other instead of effectively fight the government. However, the plan backfired terribly, and the result was that it ended creating a lot of small competing parties both in the situation as in the opposition.
So, what would be set in your fictional 1950's or 1960's US was something like that, which would lead to severe and irreversible fragmentation of the political parties. That era was marked by the anti-communist paranoia, so a way to achieve that was some sort of big scandal, much worser than Watergate, which presents severe, large and deep communist infiltration and manipulation rooted inside both of the parties. The cause of the scandal could be someone presenting or leaking convicing evidence that one of the parties was in fact controlled by Moscow, and the scandal escalates quickly enough to show that the truth is that both parties are Moscow-controlled puppets. Some top-secret information leaked from CIA or KGB printed in the first page of a NY Times edition would be able to ignite that, with things just getting worse and worse as the situation progresses. However there should still be an important big quirk/gotcha to prevent that somebody just simply say "f--- that" and hit the big red button leading to WW3.
The public opinion would then perceive that both the Democratic and the Republican parties as traitors. Even democratic and republican congressmen and senators would, with many of them leaving en masse their former parties. As a consequence, both parties quickly disintegrates into a lot of smaller parties which distrusts and infight between them. Defecting/defected deputies and senators from both of the former parties would quickly be able to approve any constitutional changes which would be need to allow an easy creation of new parties.
Those newly created parties will tend to attract people which shares very similar toughts while being hostile to people who thinks differently, even in minor points, very unlike than the present bipartidarian system where each party hosts groups with very different political views. This would quickly lead to a situation where parties have unreconciliable political views and each one of them would oppose the others in varying degrees.
As the years passes, eventually unrelated or distant related reasons would make some of those new parties go extinguished, some new ones born, some of them fuse/rejoin and some or them being further divided. And this would persist for many and many years. With some time, a party would only rarely trust other party, and when that would happen, it would be just on some pontual and occasional situations when they agree more by coincidence than anything else or just want to defeat a proposal or idea from some other third party who both consider as a greater foe.
Finally, something must set in the public and congressmen opinion the idea that multiple parties are better than just two parties because puppeting a lot of small independent parties should be much harder than puppeting just two big ones. Some strong evidence should be presented that somehow, Moscow cannot puppet America's parties efficiently anymore in this new situation and that Moscow wanted to preserve the US bipartidarism system. This is important to keep discredited the idea that fragmentation and disorganization was one of Moscow's objectives, otherwise eventually someone/something might refound the bipartidary system or just start a single-party one.
In this setting, anybody could be your activist. Somebody that told in antecipation that Moscow was controlling both parties and tried to warn America's about that months or even a few years before the scandal is revealed is a likely candidate for winning the presidency after the scandal is revealed. Your activist is much more likely to be some extreme or moderate right-wing communistphobic guy, but it is also perfectly possible that somebody else, even a left-wing (although still anti-soviet) guy could be.
The problem is that a first-past-the-post system tends to coalesce around two parties. And since everybody elected in a first-past-the-post system is elected under such rules, those are rules that work well for all of those who have the power to change the rules, which makes then uninterested to change such rules.
So, to break the tide-lock of a two-party first-past-the-post system, you would need a third party ideologically devoted to breaking the system. That would have to be a highly centralised party, able and willing to expell any members who betray the aim of ending the two-party system.
This party then would have to exploit the frail points of the FPTP system. One such is clearly the tendency to create "safe" electoral districts - in the US, "red" and "blue" States, congressional districts, etc. At some point, a State or district will become so "blue", or, conversely, so "red", that they weaker party will cease to try to win such district. You then have a district where a minoritary, but sizeable, chunk of the electorate has been abandoned by its preferred party: red voters in blue districts, blue voters in red districts. Your third, let's say "yellow" party, wants these voters, but it also wants occasionally disgruntled voters from the district major party.
There are two possible tactics here: one, you run a "purple" platform, which gets the minority voters en bloc because purple is a lesser evil to the disgruntled minority, but perhaps still acceptable for a part of the majority. Or you run a "crazy" platform that seems "redder than red" in a red district (and conversely, "bluer than blue" in a blue district), and then work to tear apart the majority, while making dog-whistle signs to the minority, in the expectation that they hate the dominant party so much that they will vote for anything that can break its hegemony. Either way, you hope to win a few districts where the majority party is overconfident and lazy, and to come up with a significant number of second places nationwide, so that you can establish yourself not as a national third party, but as a regional second party wherever possible.
If you can win a few local governments, you then try to make them poster-cases for your yellow party and yellow policies. Where you are unable to do that, you attempt to at least get a few representatives and city councelours and make them an earnest oppositional force, if possibly more earnest - more decisive and more competent - than the traditional parties.
Your yellow party then systematically strives for legislation changes that undermine the FPTP dynamics. You want proportional representation, multi-vote, double ballots in case of no majority, party lists, party discipline, no State (including no judicial branch) interference in parties internal affairs (so that you can effectively resist the tendency of your own people to get comfortable in getting elected in FPTP), term limits (to force the major parties to change candidates and occasionally run less experienced politicians), etc.
You will evidently have to find ways of financing your movement; you will have to treat donors as you treat voters, identifying those who are more concerned with defeating one of the official parties than with supporting the other.
And then you need to do this for several electoral cycles, and expect that a political or economic crisis will allow you to surpass one or both the ruling parties and allow you to implement an electoral reform that effectively changes the system. Or you then you make a U turn and accept FPTP now that it favours you, and watch the weakest of the two old parties die slowly...
Simple.
First, join either the Republican or Democratic establishments.
Second, assuming that the activist is a decent politician, gather votes at at various rallies through the country. When primaries come, this activist should either win states he rallied in or come in a respectable second. He should also show respect to other candidates from his party, and maybe even make friends with them.
Third, now that he's on top and set to clench his party's nomination, a day before the party's convention, he should drop out of his party and declare himself an independent.
Fourth, he should ask one of his fellow candidates who is still in the party but didn't get the nomination to become his vice president. Hopefully, he/she accepts. Now, a new influx of voters will come to vote for this dynamic duo.
Fifth, this activist now has almost a 1/3 chance in becoming president. The hardest part of the presidential process is getting on to the ballot. From there, with some epic advertising and rallies to stir the crowd up, your unnamed activist should be able to become President.
The problem with breaking a 2 party system is that you often need 4 parties and not 3 to change the status quo.
If a new independent party appears on the right (like the tea party) then it splits the Republican vote, meaning the Democrats take power. Which doesn't break the status quo.
What you need is for both the Democrats and the Republicans to split at the same time. In terms of narrative I think you would need a historic level decision to usher the change. A potential example of this could be the proposed Franco-British Union. What if this pact was offered to America after WW2? Would republicans be split along nationalist/imperialist lines? Would the Democrats see it as an opportunity to bring in European style socialism to America or a costly burdon?
You don't need to win the presidency to have political impact. Trying to change the voting system on the Federal level is a relatively hard problem because it requires you to convince a lot of different people.
if you wanted to fight the two party system it makes much more sense to focus on a more local level. Getting a single US state to switch to Party List Proportional Representation for it's state election would be much easier than to try to get the whole US to change.
It's quite possible... and it nearly happened. Ross Perot was competitive at points in 1992 before he dropped out of the race. I believe at one point he was actually leading. He later got back into the race, but never returned to his old polling numbers. Had he not dropped out in the first place, who knows what would have happened.
Ultimately you just need the right candidate that captures the interest of the general voting populace. And a lot of money, obviously. Its certainly a lot easier to do with the backing of a party, but not impossible without it.
