Rapidly driving a keyboard signal
Realistically the only way to achieve maximum rate is to simulate it in the keyboard implementation directly, alternating the state on each poll by the USB host.  Likely the limit is going to be 500 Hz, ie, change state on each millisecond polling, but perhaps there is a way to send multiple key events per poll, you will have to study the HID keyboard spec.

Thus to really achieve your goal, you want to find an MCU eval board with native USB and a USB keyboard example project, and "hack" it to do this.  An Arduino Leonardo, various "Teensy" and some MBED boards may have this capability.

Electrically "pressing" keys on an existing keyboard implementation would have to be a lot slower (or else queued by exfiltrating a timing trigger somehow) to avoid things like severe slowness of actual change resulting from mistaken matchups between intervals.

You may be able to do faster still by faking things in software from the hosting OS driver side...