Your body should stay loose and supple until the moment of impact, allowing for maximum acceleration into the target.  In contrast, novices tend to tense up the beginning of the kick.  The tension in their body causes them to decelerate even before they make contact.
That said, it is important to fully engage your core just before you hit your target.  If you fail to do so, you will diffuse the impact of the kick by allowing your posture to deform.  A rigid core will maximize the transfer of rotational force.
Excess tension will telegraph kicks to your opponent.  This is most often revealed in exaggerated hip movements at the beginning of the kick.  Tension also causes some people to hunch over and bend forward.  This makes it more difficult to maintain an upright and balanced posture throughout and turn over the hips to create rotational force.
Over-tension also causes you to expend unnecessary amounts of energy in the course of a fight or training session.  While good athletes can “muscle” strikes early on, they tend to slow down quickly and “gas out” in later rounds.  Efficient technical strikers can keep a high output and consistent power throughout.

Balance and Posture

“If you wish to control others you must first control yourself” ― Musashi Miyamoto
Maintaining your balance through all phases of a technique puts you in the best position to defend, course correct and counter attack.  Poor balance will leave you out of position, prolonging the recovery phase of the kick and leaving you vulnerable to counters.
To have good balance for the round kick, you must always maintain an upright posture through the technique.  If you do not keep your body straight, the rotation force of the kick will throw you off-balance whether you miss, land or get blocked.

Good balance begins from the ground up with a raised heel, a straight supporting leg, a neutral spine and a stable core.

Weight Transfer

“When torrential water tosses boulders, it is because of its momentum.” – Sun Tzu, the Art of War
While we obviously cannot change our bodyweight in the course of a fight, we can increase the amount of weight we harness in a kick with a slight forward lean.  This is similar in concept to “sitting down on your punches”  where move your bodyweight towards the target.

We will very often see beginners leaning backwards when kicking to compensate for deficits in flexibility or mistakes in the angle of the kick.  This significantly decreases the force of the kick by pulling their weight away from the target.
Others may pull their head back for fear of being punched.  However, the better way to address this is with proper footwork which allows you to angle off to the side and take your head off the center line.
To ensure good weight transfer, take an outward step to generate forward momentum at the beginning of the kick and maintain good upright posture during the follow through.

Follow Through

“When you decide to attack, keep calm and dash in quickly, forestalling the enemy…attack with a feeling of constantly crushing the enemy, from first to last.” ― Miyamoto Musashi, A Book of Five Rings

At its core, the muay thai round kick harnesses the rotational force (or torque) of your entire body.  Your foot is the pivot point and your kicking leg is the lever.  All things equal,  the greater the angle of rotation, the greater the magnitude of torque.  The more your body rotates, the harder you kick.

To maximize follow through, always think of accelerating and swinging your leg through the target.  Too often, novices will just reach the pads with their shin, pulling their leg back immediately after making contact.

When practicing round kicks while shadowboxing, develop good follow through by spinning through the kick instead of stopping your leg in mid air.  You should have enough angular momentum for a half to full turn.

Timing

“When the strike of a hawk breaks the body of its prey, it is because of timing.” – Sun Tzu, The Art of War
As a general rule, it is best to attack when your opponent in the worst position to defend.  The most common defense to a rear round kick is the knee shield.  Assuming you and your opponent are in orthodox stances, a great time to throw a right round kick is when your opponent is heavy on his lead left leg.  Putting weight on his lead leg means he will need more time to raise the left knee shield.

Another opportune time is when you opponent side steps to the left.  Not only is he heavy on his lead left foot, his body is moving towards the power arc of your right kick.  His sideways momentum will make the right round kick all the more devastating.
A right round kick is also a good interrupting counter to a left jab.  Your opponent exposes his ribs when he extends this left arm.  He will also tend be heavy on his lead left leg making it more difficult to raise the knee shield in time.  You will naturally avoid the jab with an outward step when you kick, taking your head off the center line.

Round Kick Coaching Cues

Raise and Rotate the Support Foot

It is also vital to get up on the toes of your support foot when you kick.  Raising up on the ball of your foot allows you to rotate your body more easily.   It reduces friction by minimizing the surface area of your foot.  This allows you to rotate at a greater angle and at a higher speed, both important factors for achieving a harder kick.  You will also find rotating flat footed quite difficult and will likely make you loose your balance.

Getting up on your toes will also a couple of inches of height to your kick.  If your opponent blocks, the additional height may help you get your kick above your opponent’s knee shield and into his forearm.

Forward Step

The appropriate distance and angle of the initial step with the supporting foot will depend on your opponent’s position relative to you.  At mid range (e.g. boxing / kicking), you should take a small outward step on about a 45 degree angle.  This creates a good angle for you to land your kick in your opponent’s ribs while taking your head off the center line to avoid straight counterpunches.

It is possible to land a good round kick from at fairly close range (e.g. boxing / elbow distance), however, you will need a take a even wider outward step to compensate – approximately 25-35 degrees as a general rule.  Done correctly, your shin will land in your opponent’s stomach and rather than his ribs.  A kick from this distance will likely have a bit less force as your kicking leg will rotate less before making impact.  However, it will leave you facing your opponent’s flank, in a good position to land follow on strikes.
If you take a long step to get into range, it is important to shuffle your rear foot forward so you are in a stable and balanced position before you lift your rear leg off the ground.  If you lift your kicking leg from a very wide stance, you will tend to fall forward after the kick, leaving you vulnerable to counter attack.

Pushing off the Rear Foot

Many novices rely primarily on the hip flexors to lift the kicking leg.  A common saying in the boxing world is power is generated from the ground up.  This applies to kicking as well.  In order to deliver a powerful round kick, you must push off the ground with your rear foot and harness the powerful quadricep and calf muscles on your kicking leg.

Pushing off the rear foot will also help bring your body weight upwards into your target.  It will also put you in a more upright position making it easier to maintain good stable posture throughout the technique.

Angle of Attack

The target of the muay thai round kick is your opponent’s ribs.  For an opponent of similar height and proportions, standing straight-on in mid-range, your shin will need to move along a 70-80 degree angle to directly reach his ribs if you maintain an upright posture through the kick.

Despite this, many novices move their kicking leg on a strange elliptic path.  They first tilt their upper body to the side to raise the leg, and then swing the leg towards the opponent along a horizontal plane.  These kicks often only reach the opponent’s hip height, and are usually quite easily defended.  This is a pretty common error. It may also be that people confuse the body kick and leg kick techniques.
As we discussed earlier, tilting the torso to the side puts you in an unstable position, further prolonging the recovery phase.  It also limits weight transfer, reducing the impact of the kick.  The kick also losses rotational force if the leg travels along an awkward elliptic trajectory (up and in from the side) instead of a more direct path.

Straighten the Kicking Leg

It is best to maintain a fairly straight kicking leg throughout the entire technique.  As mentioned earlier, the muay thai round kick relies on rotational force for its devastating power.  Your kicking leg is a lever which rotates around your support foot, the pivot point.  If you bend your leg during the kick, you shorten the length of the lever, thus reducing torque.  Coincidentally, this is also why you’ll see some fighters bend the leg to slow their rotation after missing the opponent with a kick.

Some will bend the leg because of limited flexibility in hamstrings, psoas and glute medius. Other may bend the kicking leg because they confuse a number of similar looking martial arts techniques with the muay thai round kick.
The karate roundhouse kick (mawashi geri)  requires that you raise your kicking leg in a bent and chambered position, then extend the lower leg in a snapping motion and kick with the ball of your foot (not the shin).
The question mark (or “Brazilian”) kick is essentially a front push kick feint which morphs into a head kick mid strike.  While it can be effective in competition, it is an entirely different technique than the muay thai round kick and relies on extreme hip turnover a downward flicking motion for its power.

Hip Thrust and Turn

“Hips don’t lie” – Shakira
Correct hip movement is vital to weight transfer and good balance during the kick.  You will find it extremely difficult to walk forward, let alone kick, with your butt tucked backwards.  You must thrust your hips forward to transfer your body weight into a kick.
A complimentary movement to the hip thrust is swinging your rear hand downwards.  This arm movement help you open your hips up and maintain balance.  Some dutch-style coaches will advise keeping both hands by your the chin when kicking, however, I notice a significant drop-off in power and balance when doing so.  If you angle off correctly with your supporting foot, I find keeping the lead hand by your chin to be sufficient.

A slight hip turn is necessary to guide your shin along its 70-80 degree path into its target.  This is however a very smooth and gradual turn.  Some people make a sharp, jerky twist with their hips to to create a sharp flicking motion at the end of the kick.  This will compromise balance and do damage your back muscles over time.
I find the hip flick to be common amongst people who either tilt too much to the side or who do not turn the hip enough.  For those who lean to the side, the movement of the kicking leg tends to lag that of the body.  And so, they require extra hip movement to reach the target with the shin.
Those who fail to turn their hips enough have their kicking leg traveling on a very vertical trajectory.  They use the hip flick to essentially course correct and bring the shin into the target.

Recovering your Fighting Stance

“In all forms of strategy, it is necessary to maintain the combat stance in everyday life and to make your everyday stance your combat stance. You must research this well. – Miyamoto Musashi”
The ability to quickly recover your fighting stance is an oft neglected aspect of round kick technique.  It is vital to defending your opponents counterattacks and launching follow on strikes.  Unbalanced novices will often fall forward into the opposite stance (e.g. kicking with the right leg from orthodox and ending in southpaw).  They be in range of their opponent’s counterattacks but may need some time to switch back to an orthodox stance before they can be fully effective.

Good balance and upright posture goes far in expediting the recovery phase of a kick.  If you make contact, immediately relax your body and return the kicking leg along the same path / trajectory.  Stiff muscles will impede recovery as much as attack.
If you miss, it usually better to swing through to face your opponent than stopping mid-strike by tensing your body.  Although you keep your eyes on your opponent by stopping the kick, it usually takes longer to recover your balance.  Even if you get countered while spinning through, you are usually in a better position to absorb damage if your posture is balanced.

Breath Control

You will find many thai fighters shout “ooh wee,” “essh” or exhale sharply while kicking.  While this helps them make their case to the judges and spectators, it also serves a physiological purpose.  You naturally tighten your core muscles when you exhale, grunt or shout.  The tightening of the core can lead to more forceful muscle contractions throughout the body.
Anecdotally, I find exhaling after each strike also helps me stay loose and relaxed and lowers the perceived exertion of pad / bag rounds.  Sharp exhalation clears the lungs of used air, allowing for deeper diaphragmatic breaths.   This in turn, has a calming effect on heart rate.
Thank you Elaine Fung for finding this article and videos. I think it will be useful for the students at the school and for Kickboxing Vancouver