I think it's surprisingly simple. Just have an increasingly partisan split between the parties. Go back to any election and get elected the lesser, more partisan choice for about 3 or 4 terms in a row. Do this in a time of economic stagnation. One side becomes so recalcitrantly conservative that they won't adopt new measures or amendments. The other pushing taxes so high that it bankrupts the populace. You have then a chance for a strong centrist candidate coming up from the middle at a time when people are so sick of the Dems and Reps that they'll try anything. Someone established and well-liked who decides to position themselves as an Independent. If he does well, next election cycle you have some Senators and Congressmen switching allegiance to this newly formed `Independent Party' and maybe you have some new Independents elected. Have them form a party platform next election. If he can get re-elected, a precedent is set.
Some points in history this might be possible:
28- Al Smith wins. Depression happens. Can't control it (was against New Deal).
32- Hoover comes in. Problematic as well.
36- FDR probably becomes nominee, but possibly not. Let's say Huey Long.
40- With World War II looming and the economy not that well off, no one wants a war. Robert Taft elected. Isolationism ensues.
44- Enter an Independent candidate who wants Internationalism.
64- Goldwater somehow wins (maybe if JFK has an unsuccessful, uneventful presidency)
68- People hate Goldwater, think he's a warmonger. Anyone will go in. Turns out it's McGovern.
72- People hate taxes more. Are willing to try Republicans again. Enter Nixon.
76- Nixon is caught at his dirty tricks. People are sick of it. Willing to support an Independent candidate who is honest, anti-war but moderate on government size.
just off the top of my head.