You wouldn't connect the Xbox directly to your computer, but yes your Xbox can stream files off your computer via Windows Media Center.
I have my Xbox connected to a switch, which is hardwired to my router.
I don't use it very often, but it seems to play most video codecs, can't remember if mkv is one of them.
I set this up for a friend. Connect both to your router. Then you'll need some kind of DLNA software on your computer. You could run Windows Media Center, but I preferred TVersity. MKV isn't a codec, but rather a container. The Xbox 360 will not play all of the codecs in this container. Your best best is to use TVersity to transcode the video file at run time. This is right in the settings. There are tons of guides online, just search Google for Xbox 360 TVersity.