PISO shift register to USB
A relatively straightforward problem: 


Create a small state machine to drive your parallel capture, serial clock and character serializer. This is very easy with a small ROM/EEPROM and a couple of multiplexers.  
Transfer 3 bits at a time into another 4 * LS165 configured as a 3 character serializer. Configure the three characters as 'group' (char 'A' to 'I'), 'Value' (char '0' to '8'), CR (chare carriage return).   
Create a baud rate generator to transfer your characters from the serializer and provide a clock for the state machine.   
Clock the serial data into an FT232H configured as a UART to provide your serial to USB conversion.  
For extra points, receive a serial command from the PC to start and stop your serializer. 


All up about 15+ chips, but easy to layout.

OR

Use something like an Arduino Nano for about $4 (including USB connectors and cable) and write a few lines of code to transfer the data. You could even use the newer Arduino Micro based on the ATMega32u4 and get a single chip solution with USB connectivity for about the same price.   

Instead of using TTL shift registers you'd be far better using something like the 16bit MCP23017 or the PCA9671 I2C expanders, both readily supported and relatively cheap.