What kind of serial cable do I need? (TTL vs RS232)
Based on the voltages being +/- 6V, you need a USB to serial cable that uses RS232 levels (not TTL) and 3.5mm plug. There are several ways you can achieve that, which might include adding RS232 tranceiver to your TTL interface,  or direct USB-RS232 adapter that has one buit it. The remaining task is to interface the right connector.