I don't think this makes sense to do with an Arduino - there's just nothing for it to do, and considering the necessary external components that would be required to control the current and voltage being applied to the battery.
For charging a battery while it's in a phone, you just supply 5v (well, 5.2 is often better to help compensate for the loss in the line - this is what most plug in chargers do, whether or not they mention it) via USB - the phone does all the work.
For charging a battery outside of a phone, they're single-cell LiPo batteries; there are charger IC's that cost well under a dollar and require nothing but an external capacitor (and, for some, an external current setting resistor of some sort), and take ~5v in (these are what all our cheap charge-through-USB devices use), and do the ramping back of the current automatically.
Also, it sounds like you have little idea how a battery is supposed to be charged or are confused as to the nature of voltage - you're supposed to charge at limited current while battery charge is very low (ie overdischarged), constant current until near full charge, then slow it down and finish with constant voltage charge.
Also, what does this have to do with wireless energy transfer or coils? Or are you trying to emulate one of the inductive charging devices?