MadDriver, how experienced are you at soldering?
It is not my specialty so if I can not do it I take it to my Local Hobby Shop and for a few dollars apiece he is always more-than-happy to do it for me. And I know it will work.
If you are in the same boat (so-to-speak) that could be your best choice if you have a Local Shop. They will be very happy to solder the ESC, battery and if necessary the charger. And he can explain exactly how to hook up the charger to the battery.

I will take a guess here but you would be better off going to your local shop.

Those loose alligator clips on the right of your charger, I can see in the pic you posted they have a connection at the ends that insert into the banana plugs coming out of your charger. Plug those together so that you have alligator clips coming out of your charger. I do not know anything about balancers so I can not help you there (don't even know what they are), but you can (as I would do) just get yourself a 12volt battery to place in your fieldbox and connect the alligator clips directly to the battery. Then you will need the Great Planes Charge Lead Banana Plugs mentioned above http://www2.towerhobbies.com/cgi-bin...&I=LXNCU4&P=ML and insert them into your output ports on the front of your charger and plug the male end into your female end on the battery to charge it.

That is the route I would personally go and omit the balancer as (like I said) I know nothing about them. And I don't see the point in attaching the charger to a car battery when all you need is a 12volt in your box.

If this is all getting too confusing for you it would be worth the few dollars apiece to go to local hobby shop and have them set it all up for you.

And you WILL need a separate battery pack if you wanna fly and charge at same time. Which kinda sucks for me as I have at this moment 3 electric models (electric cub, Gentle Lady glider, and Lanier ST-40 [which that battery pack itself is around $140]).

Keep us posted please.