The tailpiece (the long grey shaft-type thing with a notch), which attaches to the back of the plug, has to be aligned with the cutout in the plate which has the key bitting code. Once this is done, there is another plate beneath the coded plate, which has to be pressed towards the centre line of the tail piece (you'll see a tab which is covering the spring, opposite to the bitting code.....push it towards the centre line of the lock)
When the lower plate has been depressed, the tailpiece will be allowed to be removed by pulling it away from the back of the plug.
Since you have the key, insert it and turn the plug to approx 45 degrees. There are two spring tabs which lock the plug/cylinder into the knob...these need to be manipulated with a slim flat-bladed screwdriver, so as to free the cylinder. Pulling on the key bow whilst carefully prying the spring tabs will allow the cylinder to be removed.
Alternatively, you could fork out the few dollars it costs to get a Kwikset rekeying tool that will do the job properly.