External 5V Power and USB Serial I have a Chinese Arduino nano clone, but I don't know if it's possible to connect the Arduino to external (5v) power and have a USB serial connection ongoing. I'm trying to make a PC controlled LED Strip that can also work on its own.

For example: the PC controls the Arduino sets it to a specific mode and I turn off the PC, but the Arduino should still be running even when the PC is turned off.

I'm new to the world of arduinos.

I'm planning to use this to control addressable LED RGB Strips and am planning to use a 4000mA max power supply.
Having used counterless Arduinos and clones, I can tell you that you can connect both without worrying about any danger. Unless you have a very faulty board, your pc won't be damaged, almost surely neither will the board, as is should select the most powerful source, between your usb or a battery/external PS, while leaving untouched the serial line.