Connect ESP-01 to USB Uart
Just to give an anwser for users struggling with the same problem I will sum up what helped me.

To connect the UART to the ESP the Beginner's Guide to the ESP8266 from @tttapa was very helpful. Another helpful link was this one:
https://www.allaboutcircuits.com/projects/flashing-the-ESP-01-firmware-to-SDK-v2.0.0-is-easier-now/

So in my case I just needed the GND, 3V3, RXI and TX0 pins from my UART.

I needed no external powersupply for the flashing process and some simple examples.