A "full string" would be multiple characters stored after each other in memory. Either with a special terminating character at the end, or with a length stored somewhere.

The serial interface processes data one character at a time.

So wouldn't the most obvious approach be to implement a loop that iterates "len" times (if the string is stored with a length value) or until a magic end-of-string character is found, and sends found characters to the serial port after having made sure that the serial port is ready to accept more data?

So where are you stuck?