Kurtis Bowler of Rainier CrossFit is a CrossFitter with an affinity for strongman movements, and in this video he goes over the various ways to flip a tire.
You can take a wide snatch grip or a narrow sumo grip on the tire, and with lighter tires athletes can choose to drive it all the way up in one movement (much like a snatch or power clean). With heavier rubber, athletes will be forced to add in a few more steps. The best strategy is drive into the tire, step forward and extend the hips. At this point you should drive off one leg while kicking into the tire with the other. The sumo grip will also involve more of a shrug to elevate the tire.
Common faults involve deadlifting the tire rather than driving into it and setting your hips in a poor starting position.
Additional video: Strongman Hip Extension by Rob Orlando, published Dec. 9, 2009.