This is a scenario that I have recently made up: A spacecraft is at the edge of the solar system, in orbit around Pluto, to make its journey to Alpha Centauri. The spacecraft wouldn't accelerate from Pluto to Alpha Centauri, it would go all the way back down through the solar system, doing gravitational slingshots from all of the planets and ultimately the Sun, assuming that all of the planets were in the correct alignment. Would this actually help the spacecraft, and would it take an amount of time that would be reasonable?
Unless I am mistaken, the amount of speed a gravitational slingshot can add is less than twice the orbital speed of the planet being used. Mercury's orbital speed is the fastest, at a little over 47 km/sec. If you could use each planet once, and each one gave you a little over twice Mercury's orbital speed (which they would not even come close to), then you would gain 800 km/sec, which is a lot. But it would still take you over 1600 years to get to Alpha Centauri.
And this is overly optimistic, perhaps by a factor of 10 or more. Adding double the speed of a planet is based on a simplified model of a ping-pong ball bouncing off the front of a semi truck, which depends on the ball going out exactly the opposite direction it came in. But actually in space the object swings behind the planet, and the faster it is going the less time it spends close to the planet and the less additional speed it can pick up. Once you are at solar system escape velocity you can't swing by repeatedly to pick up more speed, either.
Interstellar travel is just hard.
Added later: The accepted answer to this question looks at what you can accomplish with gravity assists and other techniques. It doesn't mention the specific approach you proposed, but that answer gives a sense for what can be accomplished using the best combination of maneuvers.