Understand what a punch is. The power of a punch does not come from your hand or arm. To punch hard you must punch with your entire body. This involves your legs, your hips, your shoulders, your arm and finally your fist.
Start with your feet. Stand with your legs comfortably shoulder width apart then step forward with the foot on the side of your non punching arm. If you are right handed step forward with your left foot. Raise your arms in front of you in punching position.
Crank your hips as you load up for the punch. As you throw the punch your hips will turn. This is the power in your punch. The more your hips whip around the more powerful your punch.
Cock your shoulders for the punch. Twist them back like your hips and draw your punching arm back. Tighten your punching hand into a tight fist. The tighter the better. Be sure that your thumb is outside of your fingers.
Eye your target. Do not guess at your impact point. You should zero in on a specific spot where you want your punch to land. Focus is important. If you punch wildly you could hurt yourself.
Unleash your punch. It starts from your feet. They should be solidly planted on the ground. Power should launch from your back foot. Push off the ground as you rotate your hips and shoulders toward your target. Your arm then extends forward in a straight line from your shoulder. Timing is key. Your fist should meet its target at the precise moment your arm, shoulders and hips are fully extended. The power should be from the ground, up your leg, through your hips, shoulders, arm and into your fist. All your energy is now transferred into your target. Be sure to keep your wrist straight and hit the mark with the two knuckles closest to your thumb. They are the toughest bones in your hand.
Practice, practice, practice. Good punching technique comes from constant practice. You must develop muscle memory of what a perfect punch feels like. That is the only way to teach your body to do it. Hit a punching bag while focusing on perfect form. As you feel more comfortable increase the intensity and speed.
Get stronger. Once you have perfect punching technique you can work on making yourself stronger. A perfect punch involves most of your body's muscles so a full body workout is beneficial. Your back and hips are especially important. Any strength exercises that emphasize your core will help you punch harder.