In balancing cat pose the same side arm and leg are both lifted.
One of the first things that you can practice, even before trying to balance, is spreading the shoulder blades to lift the ribcage.
This action uses the serratus anterior muscle. When doing this action, you may find that your arm and shoulder feels more stable if you rotate the arms outwards. You can also try squeezing the elbow joint. If you aren't sure, or are having difficulty, practice lifting and lowering your ribcage. Note that I am not bending my spine backwards as I do this in the pictures below. I'm simply moving my ribcage relative to my shoulders.
The idea is to use this same action and keep it when supporting your weight on only one arm.
You can learn balancing cat pose in steps.
The first step is to lift the arm. On all fours:
When shifting your ribcage focus on feeling your hands. Notice when all of your weight is on one hand. The other arm may feel relaxed. Then you can lift that arm.
Make your neck feel long and reach that arm forwards.
Then relax and repeat.
Next with ribcage lifted at the shoulders,
When shifting your pelvis focus on feeling your knees. Notice when the weight of your pelvis is entirely over one knee. Then lift your other leg. Make your leg and spine feel long.
Then relax and repeat.
Next put these elements together for a complete balance exercise.
Focus on reaching forwards with the arm and back with the leg. Then release and repeat on the other side.
When I first started doing this pose I used to lift the leg first. But then I found it was easier to stay balanced when I lifted the arm first. Once you are used to this yoga pose you can try both options.
You can also experiment with shifting your weight forwards or backwards slightly. I find that when I shift my weight back the top of my back foot presses into the floor with more pressure. I can then use this foot to help make balancing cat pose easier.
Once you are used to balancing focus on making your arm, leg and spine long. And work at keeping your torso level from left to right so that the lifted limb side or your ribcage and pelvis is at the same height as your supporting limb side.
Note that their are also downward dog and plank variations of this yoga balance exercise.
Although this is mainly a balancing yoga pose, you can also use this yoga pose to strengthen the back of your body. Focus on using the hamstrings and glute of your lifted leg to lift the leg higher. And focus on using your spinal erectors to bend your spine backwards (and tilt your pelvis forwards) at the same time.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.