USB and Arduino: Power Arduino, charge LiPo, and transfer data at the same time

I'd recommend using the Arduino Leonardo as a reference for your circuit. Its pretty much the same as an Arduino UNO but has a built in USB interface so you don't need the USB-Serial converter.

I'd also recommend running the chip at 3.3V. That way you can use a linear regulator to regulate the 3.7V battery voltage down. Otherwise you'd need a boost converter to bump the voltage up to 5V which needs more parts.

If you follow those recommendations then all you will need is a LiPo charging IC to handle the charging. You'll need to add an ISCP header to flash the initial Leonardo bootloader, but after that you can use USB and the Arduino IDE to program it.

I did exactly this for a school project a while back. You can find the schematics here: https://github.com/seanwatson/ece4416-project/blob/master/hardware/rev2/sw-4416-project-v2-sch.pdf

Use a TP4056 1A LiPo charger board (it has overdischarge protection) and a boost converter, like the XL6009. It will run you at most 5$ if you are prudent. Overdischarge protection is important.

If you want a port for programming, add a micro-USB breakout board, and if necessary a pushbutton for bringing RST down to GND. I prefer just having GND, TX, RX pins on a 3 pin header, and plugging my cable into that.