Do you plant both heels before throwing the next punch?

No, I lean forward, and rise up on the ball of my back foot, and throw my weight into it with a bounce as I push off like am taking a false step. This way I can recoil quickly to double up on my punch.

 

He doesn't explain it well.
Punching starts from the floor up and you turn your weight before you move your hands. You are trying to whip your fist through the target, not push it through.

 

This thread belongs in the Training section.

 

You are probably doing it wrong, and the guy in the video doesn't explain it well. I could show you how to do it and you would notice the difference in 10-15 minutes but trying to type it....
Look up a book called Boxing by Edwin L Haislet and read his instructions on how to throw the hook and the right hand and work through them. His instructions will get you turning your hips ahead of your hands.
There are two common mistakes you see in punching. One is punching from the top down, when you swing your arms hard enough to make you shift the weight on your feet. The other is forgetting that a combination is made up of individual punches.

 

No.

 

The heel isn't the X factor here. The hip is.

If you want to throw a punch with your right hand you have to use your right hip and if you want to throw another punch with your left hand you have to use the left hip.

Moving up your heels is just the consequence of extending your hip so you can generate power and length. If you can throw a proper single right hook and a proper single left hook you can also throw a proper right hook left hook combo. If not...train the single punches because you're clearly not doing it right.

1. Hip extends to generate power and length
2. Heel rises as a consequence of extending your hip
3. Punch is thrown

As soon as your hand moves back to your head to protect you, your hip also goes back so goes your heel down while you're extending the other hip to follow up with another punch. And then you have the pattern from above. So you have no time in between where you planting any feet.

 

The Heel is your Bio mechanical Break.

 

I punch strangely, but very powerfully. I step into all of my punches, most especially the jab and straight right/right cross. What I mean is, I launch the punch (for instance) with my right fist going forward while twisting my waist into it, my right leg stepping forward into the punch as well. Sounds weird (it is weird), but for my body mechanics this way the punch has the whole side of my body going into it.

The main problem is getting my feet back into line.. I usually have to bend my front left leg and throw the hook right behind the right. All the while my back leg is getting back into position and I'm ready to, say, throw a jab.

 

That's pretty much what you are supposed to do.

 

I don't really see it too often, or perhaps most fighters are more subtle about it. You can see Shavers, Foreman, Tyson (especially the hook) doing it.

 

Stepping into a right hand can result in a devastating amount of power.

 

You don't see it often because most fighters step in with the jab but don't move their right foot forward as well. You move on, you move the other. If you do, the hook behind the right gets you back to your base. If not, the hook wrecks your balance.