Leach circuit to power microcontroller
It's a valid plan and I've seen others implement it before. Here's a link to a evaluation board that does AC lines monitoring and powers itself off of the AC Mains. Look at the schematic on page 10. The regulator is U2 and the rectification diode is Z1. You might be interested in the board anyways, it already does the voltage and current measurements and outputs the value over UART/I2C/SPI.

Back to your idea though. I would suggest the following modifications. We'll talk about the clamp in the next paragraph. Instead of a shunt capacitor make it a series capacitor. This provides AC coupling and blocks any DC current from forcing the diode on. The diode is a cheap form of rectification. Half wave rectification is probably good enough for your purposes. See the wiki page for some details on rectification. 

Make sure that the regulator can support the output voltage of the rectifier. If you bring in the voltage directly from the line, the rectified voltage will be very large. This is one of the reasons I would use a transformer to step down the voltage. Additionally, placing a shunt zener diode can help decrease the rectifier output voltage. You might want this if the rectifier output is too larger for your regulator. See what they do in the schematic linked above. U7 is the shunt zener. Make sure that you add appropriate resistance to prevent excessive current flow if the zener conducts.

One issue I see with the plan is the current transformer that you are planning to use. Current transformers are typically used for current measurements. If you look at the I-V curve in the datasheet that you've linked, it provides mV of output voltage for tens of Amps on the primary side. This cannot supply the rectifier and regulator. Alternatives would be to use a transformer to reduce the AC voltage to something manageable and use the same rectification scheme. Another method would be to attach directly to the AC line as in the board I linked to. Be very sure that all of your parts are rated for the expected voltage or current.