Repeat this pattern until you do 10 reps on the last set of push-ups. So, you'll end up doing a total of 55 push-ups, standing up between sets, and trying not to rest. Or barf.
Use a full range of motion on the push-ups. None of that halfway stuff. Most people "hit a wall" around the 7th or 8th set, so don't get too cocky when set 6 feels easy. Pride comes before the fall. If you have to rest more than a few seconds between sets, you fail. Try again another time.
Take the Texas Push-Up Challenge on an off day, or do it first thing on the day you train chest or upper body. You may want to truncate your chest workout a bit though since you'll be doing 55 intense reps of push-ups at the beginning of your normal workout.