How to achieve the highest possible apoapsis from Kerbol?

[SunlitZelkova]

I know I need to flyby Jool, but what else needs to be done? Should be periapsis during the flyby be as low as possible? Would conducting a burn during the flyby help or not? I want to launch a crewed expedition into interstellar (or the closest thing possible to interstellar) space. Of course, if there is another option that does not involve flying by Jool, I would use it too.

I am playing completely stock, with the Making History DLC.

[Superfluous J]

You don't need to flyby Jool. You just need to burn until your "orbit" around Sun is not an ellipse anymore, and then turn around and with VERY light taps, burn back until it's an ellipse. Once your Ap is past Jool it takes less and less dV (at your Sun Pe which is hopefully also in LKO).

If you DO want to do a flyby of Jool then yes lower is better, and if the encounter itself doesn't give you a hyperbolic Sun orbit then you should do the burn at Jool's Pe. But again, that is 100% not necessary and may even cost more fuel for all I know.

Note, though, that - mathematically speaking - the "highest possible" Ap over Sun is infinity. Practically speaking you'll at some point not be able to expend a low enough puff of exhaust to not send your Ap from "really high but not yet infinite" to "oops now we're hyperbolic again"