Charging circuits wirelessly
It's not worth inventing your own.  You want to look at Qi (pronounced "Chee") wireless charging, which has a standard protocol.  Charging not only involves power transfer between the transmitter and receiver, but also a wireless communication back and forth between the two.

I recently successfully designed wireless charging into a product using a bq51050B High-Efficiency Qi Wireless Power Receiver and Battery Charger which is available on Digi-Key for $4.32 in single quantities. It is designed to charge batteries like a 3.7v Li-Poly that are used in cell phones.

The only gotcha is that it is available in only VQFN and BGA packages, which are not very hobbyist friendly.  There are breakout boards, like this one, which will convert from a QFN-20 to a DIP layout, bit it costs twice as much as the chip!

The advantage to using a standard receiver chip is that you don't need to build a transmitter; they are available fairly cheaply from places like Amazon.  I used this one, which costs $13.