The Oizuki is a punch performed from Zenkutsu Dachi, with the leading hand. There seem to be two schools of thought about how to perform the Oizuki, depending upon which branch of Shotokan is practiced.
In JKA style Shotokan, the hips and shoulders are squared forward to Irimi (or forward facing) position. The lead arm is extended slightly below shoulder height with the knuckles facing downward. The rear arm is pulled back to the hip with the knuckles facing upwards. The back and neck should be straight, and the head should be held level.
The punch is performed by sliding the rear foot forward in an inward arc which brings the rear foot close to the front foot, and continues past to stop in Zenkutsu Dachi once again. The knees should remain bent throughout the movement, and care should be taken to assure the back remains straight. As the foot slides forward and out, the hands move simultaneously. The rear hand pushes forward, scraping the inside of the arm against the side of the body as it goes past. The knuckles remain facing upwards until the instant before full extension of the arm, when the fist suddenly and forcefully rotates over. The hand that was forward pulls back as the other hand moves forward, also scraping the side of the body as it is pulled back. The knuckles remain facing downwards until the moment just before the hand is fully retracted, when it also suddenly and forcefully rotates over. The rotation of the hands and setting of the hips should be timed to happen at the same time that the movement of the foot stops.
The other method of performing the Oizuki is from Shotokai style Shotokan. In this method, the hips are not held in Irimi, but in Hanmi (half forward position. In this style of hip rotation, the hips are not rotated forward, but rather rearward to a 45 degree angle similar to when performing the Gedan Barai.