Since there only seems to be a single entry on the leaderboard as of now, I decided to have a go at this myself.
I decided to go with a falcon-9-style rocket. As payload I used a few empty ore tanks, together exactly 1000kg.
So here are the stats:
Mass of payload: 1.000kg
Cost of payload: 2.188 funds
Cost of transfer stage: 2.679 funds
Cost of filled booster stage: 40.576 funds
Value of empty, landed booster (assuming empty tanks): 37.862 funds
Calculation (payload not included; Booster recovered with empty tanks): 2679+40576-37862=5393
This means the payload delivery cost for this rocket is: 5.393 funds/ton
Now to the screenshots:
Here are the two full imgur albums: VAB, Flight documentation
If you don't want to use them, here are some individual pictures since the imgur album embed is broken once again:
VAB views:
Screenshots of the flight:
According to the challenge rules you only have to intercept jool, not orbit it, so that's all I did here. Doesn't seems as clean to do as performing a proper orbit, but I completed the challenge.
I guess there are still a few possibilities to reduce the cost/ton here. You could for example use a cheaper probe core (but loose reaction wheels), remove a few struts (but loose launch stability), have a bit less fuel in the upper stage (but reduce safety margins), ditch the battery (but risk probe blackout) or use a cheaper engine on upper stage (but reduce time you have left to land the booster). Definitely happy with this design, though.
Or, if you're really insane, you could do a BFR-type rocket, basically using a recoverable booster stage to get you to orbit, then transport the payload to jool with the second stage which you also return to kerbin. Uuh, maybe not.