Transmitting data over inductive charging coils (or boosting RFID/NFC power supply)
If your data rate is low, a very simple possible solution is send the data serially by modulating the power transmitter off and on. You will need to have a capacitor on the receiver that is large enough to hold up the operation of the receiver for the duration of the longest "off" bit, but you get to pick how long this is and so you can make it relatively short (on the order of milliseconds would work if your carrier frequency is 125KHz). You also get to pick the spacing between the bits to ensure that the duty cycle is high enough to power the receiver continuously even when sending lots of off bits. 

The data/power receiver circuit could be as simple as....







simulate this circuit &ndash; Schematic created using CircuitLab

There are lots of strategies to detct the incoming bits on the MCU, averaging lots of samples over time spans on the order of the size of the 0 bits (which should be very long compared to the timespans of the carrier frequency). 

Notes:


For a robust design, you will probably want some kind of voltage regulation just after the rectifier, or you can just be very careful about the coil alignment and geometry so that you can physically never have an over voltage. 
If your power requirements are large you can use full wave rectification to double output power at the cost of 2 more diodes.
For simpler code, you might want to implement a serial protocol with 2-bit symbols so a single off "blink" might mean a 1 bit and an off "blink" followed by another a specified time later would be a 0 bit. This could make decoding the stream as simple as "wait for a blink, then delay 1 time period, then look at the data level to get the received bit". 
Depending on the frequencies, coil sizes, geometry, and transmitter design, you might want to add a capacitor in parallel with the receiver loop to bring it into resonance for better efficiency.  
If you are bold, you might want to consider capacitive power transfer! It can be simpler and cheaper depending on your design (especially good if you have pieces that are tall relative to the size of the base, and potentially less alignment sensitive than coils so the pieces do not need to be so carefully aligned).