Since you are writing your logs on the SD card you should already know how to write (and read) from it. My understanding is that you have a USB connection. So, in principle, you may transfer the file by repeatedly reading the file and calling serial.write(). The process would be started by the PC sending some message to the Arduino over the serial port.
A basic "protocol" could be:
- send a command (e.g., GET) from the PC to start the transmission (no need to specify a file name, the PC will save the data under a local file name of your choice);
- answer by sending your data divided in small binary chunks (same length, may even contain a checksum);
- for each chunk of data sent, wait for the PC to send an ACK (and define what to do if no ACK is received within a certain amount of time);
- send an EOF when you have sent the last chunk of data.
This would be pure serial transmission so you shouldn't interpolate the usual debugging messages.