How should I build this circuit?
The voltage across each resistor/LED pair will be the same, so assuming that each LED has the same forward voltage, you will need to use the same value resistor regardless of how many resistor/LED pairs there are. Therefore you should use 68-ohm resistors.

That much voltage won't even be noticeable through dry skin, and copper wire is an acceptable conductor.

(Also, you don't want that wire all the way on the right side of your new schematic; that will cause a short and possibly break something.)