The behaviour of L LED depends on the code that you loaded. If the RX LED is flashing without you sending data, I suspect that the serial-to-usb converter is damaged. What is your project about?
First things to check is lsusb and dmesg (or dmesg -w) to see what the operating system thinks of the board.
Although it's not very obvious, avrdude, stk500, bootloader issues is the more suitable one; hence topic moved.