Using Arduino as Keylogger
USB-host is hard if not impossible to implement on an UNO. 

Though I believe most USB keyboards support PS2 as well (normally done with a USB-to-PS2 plug). So you should be able to talk to the keyboard using the PS2 protocol, which is a lot easier. There are libraries to do this for you. 

Next you could use V-USB to emulate a keyboard to the PC. For that USB connection you need to add some resistors and zener diodes, to stay withing the usb specification. This is all explained on the V-USB website.

Using a different Arduino that has real USB support might be a better option. Also the UNO is rather large, so not that easy to hide.