First of all, since you don't write what are you trying to achieve, let's make this clear: you can flash TMK firmware to an arduino uno, but Uno will not appear as a keyboard to the computer - atmega328p lacks USB connectivity, and on an Uno board it is connected to the computer via another "USB-to-serial" chip.
If you still want to proceed, compile the TMK firmware, and then you can flash the resulting hex file to arduino using avrdude. Here's a guide:
https://typeunsafe.wordpress.com/2011/07/22/programming-arduino-with-avrdude/There is a way to have an atmega328p pretend to be a USB keyboard to the computer, but it requires the bare m328p chip to be connected to USB using a different circuitry than the one that's present on an Uno. Furthermore, the USB protocol will be only bit-banged (i.e. no hardware support on m328p's side), so it will only support USB 1.1 (which means, for instance, that the mouse keys won't work in TMK). If you want to see the minimal circuit, have a look here:
http://matrixstorm.com/avr/tinyusbboard/Ad mchck: that one uses a Freescale ARM chip, a very different beast from AVR chips.
EDIT: You probably want to consider USB capable AVR chips for starters, e.g. (most commonly used for custom keyboards): atmega32u4 . That one's on Arduino leonardo, and more importantly on teensy 2.0:
https://www.pjrc.com/store/teensy.html Teensy is used in custom keyboards a lot.